# Full text of "Hyperbolic functions"

## See other formats

QA 342 M27 1906 i ! i ill jiiiij ! : ; ! ! : ! : ; ' i ' i! i ! i!( !' Mi i ! ! I :M i ! i ::iM | ' - ;K in I | I j II * ' i '. ISM; in ::'.': is; ;in,,Mj 5 'l!lli' ! i<"!'''!i ! '!|! H'lii- ;' . I ! i !p i i ! ! { i i ! I > |- !: i]:jiii!it(ii{!;jii;!!i;!i!iiii ill 1 lli ! '' MATHEMATICAL MONOGRAPHS EDITED BY Mansfield Merriman and Robert S. Woodward. Octavo, Cloth. No. 1. History of Modern Mathematics. By DAVID EUGENE SMITH. $1.00 net, No. 2. Synthetic Projective Geometry. By GEORGE BRUCE HALSTED. $1.00 net. No. 3. Determinants. By LAENAS GIFFORD WELD. $1.00 net. No. 4. Hyperbolic Functions. By JAMES Mc- MAHON. $i.oonet. No. 5. Harmonic Functions. By WILLIAM E. BYERLY. $1.00 net. No. 6. Qrassmann's Space Analysis. By EDWARD W. HYDE. $1.00 net. No. 7. Probability and Theory of Errors. By ROBERT S. WOODWARD. $1.00 net. No. 8. Vector Analysis and Quaternions. By ALEXANDER MACFARLANE. $1.00 net. No. 9. Differential Equations. BY WILLIAM WOOLSEY JOHNSON. $1.00 net. No. 10. The Solution of Equations. By MANSFIELD MERRIMAN. $1.00 net. No. 11. Functions of a Complex Variable. By THOMAS S. FISKE. $1.00 net. No. 12. The Theory of Relativity. By ROBERT D. CARMICHAEL. $1.00 net. No. 13. The Theory of Numbers. By ROBERT D. CARMICHAEL. fi.oo net. No. 14. Algebraic Invariants. By LEONARD E. DICKSON. $1.25 net. No. IS. Mortality Laws and Statistics. By ROBERT HENDERSON. $1.25 net. No. 16. Diophantine Analysis. By ROBERT D. CARMICHAEL. $1.25 net. No. 17. Ten British Mathematicians. By ALEX- ANDER MACFARLANE. $1.25 net. PUBLISHED BY JOHN WILEY & SONS, Inc., NEW YORK. CHAPMAN & HALL, Limited, LONDON MATHEMATICAL MONOGRAPHS. EDITED BY MANSFIELD MERRIMAN AND ROBERT S. WOODWARD. No. 4. HYPERBOLIC FUNCTIONS BY JAMES McMAHON, PROFESSOR OF MATHEMATICS IN CORNELL UNIVERSITY. FOURTH EDITION, ENLARGED. FIRST THOUSAND. NEW YORK: JOHN WILEY & SONS. LONDON: CHAPMAN & HALL, LIMITED. 1906. COPYRIGHT, 1896, BY MANSFIELD MKRRIMAN AND ROBERT S. WOODWARD UNDER THE TITLE HIGHER MATHEMATICS. First Edition, September, 1896. Second Edition, January, 1898. Third Edition, August, 1900. Fourth Edition, January, 1906. UOBKRT DRUMMOND, PWTNTPT?, EDITORS' PREFACE. THE volume called Higher Mathematics, the first edition of which was published in 1896, contained eleven chapters by eleven authors, each chapter being independent of the others, but all supposing the reader to have at least a mathematical training equivalent to that given in classical and engineering colleges. The publication of that volume is now discontinued and the chapters are issued in separate form. In these reissues it will generally be found that the monographs are enlarged by additional articles or appendices which either amplify the former presentation or record recent advances. This plan of publication has been arranged in order to meet the demand of teachers and the convenience of classes, but it is also thought that it may prove advantageous to readers in special lines of mathematical literature. It is the intention of the publishers and editors to add other monographs to the series from time to time, if the call for the same seems to warrant it. Among the topics which are under consideration are those of elliptic functions, the theory of num- bers, the group theory, the calculus of variations, and non- Euclidean geometry; possibly also monographs on branches of astronomy, mechanics, and mathematical physics may be included. It is the hope of the editors that this form of publication may tend to promote mathematical study and research over a wider field than that which the former volume has occupied. December, 1905. 111 AUTHOR'S PREFACE. This compendium of hyperbolic trigonometry was first published as a chapter in Merriman and Woodward's Higher Mathematics. There is reason to believe that it supplies a need, being adapted to two or three different types of readers. College students who have had elementary courses in trigonometry, analytic geometry, and differ- ential and integral calculus, and who wish to know something cf the hyperbolic trigonometry on account of its important and historic rela- tions to each of those branches, will, it is hoped, find these relations presented in a simple and comprehensive way in the first half of the work. Readers who have some interest in imaginaries are then intro- diiced to the more general trigonometry of the complex plane, where the circular and hyperbolic functions merge into one class of transcend- ents, the singly periodic functions, having either a real or a pure imag- inary period. For those who also wish to view the subject in some of its practical relations, numerous applications have been selected so as to illustrate the various parts of the theory, and to show its use to the physicist and engineer, appropriate numerical tables being supplied for these purposes. With all these things in mind, much thought has been given to the mode of approaching the subject, and to the presentation of funda- mental notions, and it is hoped that some improvements are discerni- ble. For instance, it has been customary to define the hyperbolic functions in relation to a sector of the rectangular hyperbola, and to take the initial radius of the sector coincident with the principal radius of the curve; in the present work, these and similar restrictions are discarded in the interest of analogy and generality, with a gain in sym- metry and simplicity, and the functions are defined as certain charac- teristic ratios belonging to any sector of any hyperbola. Such defini- tions, in connection with the fruitful notion of correspondence of points on comes, lead to simple and general proofs of the addition-theorems, from which easily follow the conversion-formulas, the derivatives, the Maclaurin expansions, and the exponential expressions. The proofs are so arranged as to apply equally to the circular functions, regarded as the characteristic ratios belonging to any elliptic sector. For those, however, who may wish to start with the exponential expressions as the definitions of the hyperbolic functions, the appropriate order of procedure is indicated on page 25, and a direct mode of bringing such exponential definitions into geometrical relation with the hyperbolic sector is shown in the Appendix. December, 1905. CONTENTS. ART. i. CORRESPONDENCE OF POINTS ON CONICS Page 7 2. AREAS OF CORRESPONDING TRIANGLES 9 3. AREAS OF CORRESPONDING SECTORS 9- 4. CHARACTERISTIC RATIOS OF SECTORIAL MEASURES 10 5. RATIOS EXPRESSED AS TRIANGLE-MEASURES 10 6. FUNCTIONAL RELATIONS FOR ELLIPSE n 7. FUNCTIONAL RELATIONS FOR HYPERBOLA n 8. RELATIONS BETWEEN HYPERBOLIC FUNCTIONS 12 9. VARIATIONS OF THE HYPERBOLIC FUNCTIONS 14 10. ANTI-HYPERBOLIC FUNCTIONS 16 11. FUNCTIONS OF SUMS AND DIFFERENCES 16 12. CONVERSION FORMULAS .'..*... 18 13. LIMITING RATIOS 19 14. DERIVATIVES OF HYPERBOLIC FUNCTIONS 20 15. DERIVATIVES OF ANTI-HYPERBOLIC FUNCTIONS 22 1 6. EXPANSION OF HYPERBOLIC FUNCTIONS 23 17. EXPONENTIAL EXPRESSIONS 24 18. EXPANSION OF ANTI-FUNCTIONS 25 19. LOGARITHMIC EXPRESSION OF ANTI-FUNCTIONS 27 20. THE GUDERMANIAN FUNCTION 28 21. CIRCULAR FUNCTIONS OF GUDERMANIAN 28 22. GUDERMANIAN ANGLE 29 23. DERIVATIVES OF GUDERMANIAN AND INVERSE 30 24. SERIES FOR GUDERMANIAN AND ITS INVERSE 31 25. GRAPHS OF HYPERBOLIC FUNCTIONS 32 26. ELEMENTARY INTEGRALS ... 35 27. FUNCTIONS OF COMPLEX NUMBERS . 38 28. ADDITION THEOREMS FOR COMPLEXES ......... 40 29. FUNCTIONS OF PURE IMAG IN ARIES . , , . 41 30. FUNCTIONS OF x+iy IN THE FORM Xi-iY , . . . . . . 43 31. THE CATENARY . , 47 32. THE CATENARY OF UNIFORM STRENGTH 49 33. THE ELASTIC CATENARY 50 34. THE TRACTORY 51 35. THE LOXODROME , 52 6 CONTENTS. ART. 36 COMBINED FLEXURE AND TENSION 53 37. ALTERNATING CURRENTS 55 38. MISCELLANEOUS APPLICATIONS 60 39. EXPLANATION OF TABLES 62 TABLE I. HYPERBOLIC FUNCTIONS 64 II. VALUES OF COSH (x+iy) AND SINH (x+ iy) 66 III. VALUES OF gdu AND 6 > 7 IV. VALUES OF gdw, LOG SINH w, LOG COSH u '7 APPENDIX. HISTORICAL AND BIBLIOGRAPHICAL 71 EXPONENTIAL EXPRESSIONS AS DEFINITIONS , 72 INDEX 73 HYPERBOLIC FUNCTIONS. ART. 1. CORRESPONDENCE OF POINTS ON CONICS. To prepare the way for a general treatment of the hyper- bolic functions a preliminary discussion is given on the relations between hyperbolic sectors. The method adopted is such as to apply at the same time to sectors of the ellipse, including the circle; and the analogy of the hyperbolic and circular functions will be obvious at every step, since the same set of equations can be read in connection with either the hyperbola or the ellipse.* It is convenient to begin with the theory of correspondence of points on two central conies of like species, i.e. either both ellipses or both hyperbolas. To obtain a definition of corresponding points, let 0^,, O l B l be conjugate radii of a central conic, and O^A^ , O^B^ conjugate radii of any other central conic of the same species; let P lt P^ be two points on the curves; and let their coordi- nates referred to the respective pairs of conjugate directions be (x^ , jj/,), (x^ , 7,); then, by analytic geometry, *2- y L - I *" *" - I (i\ i? *,'" ." V" * The hyperbolic functions are not so named on account of any analogy with what are termed Elliptic Functions. " The elliptic integrals, and thence the elliptic functions, derive their name from the early attempts of mathemati- cians at the rectification of the ellipse. ... To a certain extent this is a disadvantage; . . . because we employ the name hyperbolic function to de- note cosh u. sinh u, etc., by analogy with which the elliptic functions would be merely the circular functions cos (p, sin </>, etc. . . ."' (Greenhill, Elliptic Functions, p. 175.) tf HYPERBOLIC FUNCTIONS. Now if the points P l , P, be so situated that a t a 9 ' b, ~b^ the equalities referring to sign as well as magnitude, then P l , P t are called corresponding points in the two systems. If Q l , Q t be another pair of correspondents, then the sector and tri- angle P 1 1 Q 1 are said to correspond respectively with the sector and triangle P^O^Q^. These definitions will apply also when the conies coincide, the points P l , P a being then referred to any two pairs of conjugate diameters of the same conic. In discussing the relations between corresponding areas it is convenient to adopt the following use of the word " measure": The measure of any area connected with a given central conic is the ratio which it bears to the constant area of the triangle formed by two conjugate diameters of the same conic. I r or example, the measure of the sector A^O^P^ is the ratio sector A^O.P^ triangle A 1 O 1 B 1 AREAS OF CORRESPONDING SECTORS. ;and is to be regarded as positive or negative according as A 1 O 1 P 1 and A 1 O 1 B 1 are at the same or opposite sides of their common initial line. ART. 2. AREAS OF CORRESPONDING TRIANGLES. The areas of corresponding triangles have equal measures. For, let the coordinates of /,, Q l be (>,,/,), (#,',.?,') and ^ et those of their correspondents/^, <2 2 be (> 2 , jj/ 2 ), (x*,y)\ let the triangles P.O.Q, , P^O^ be T lt T t , and let the measuring tri- angles A^O^B^ A^O^B^ be K l , K^, and their angles <, , GO, ; then, by analytic geometry, taking account of both magnitude and direction of angles, areas, and lines, T\ = iCr.j'/ -*,>,) sin QJ = 5 ZL - fi.' A; ^j #,#! sin GD^ a l , a l , ' sn ^a i ^L _ A sin Therefore, by (2), - = -. (3) ART. 3. AREAS OF CORRESPONDING SECTORS. The areas of corresponding sectors have equal measures. For conceive the sectors S,, S 2 divided up into infinitesimal corresponding sectors ; then the respective infinitesimal corre- sponding triangles have equal measures (Art. 2) ; but the given sectors are the limits of the sums of these infinitesimal triangles, hence In particular, the sectors A.O.P^ AflJP^ have equal meas- ures ; for the initial points A lt A t are corresponding points. It may be proved conversely by an obvious reductio ad absurdum that if the initial points of two equal-measured sectors correspond, then their terminal points correspond. Thus if any radii O V A^ O^A^ be the initial lines of two equal-measured sectors whose terminal radii are O l P lt Of^ 10 HYPERBOLIC FUNCTIONS. then P lt P 9 are corresponding points referred respectively to the pairs of conjugate directions O^A lf O^B^ and 0.,A^ OJ3^\ that is, Prob. i. Prove that the sector P^O^Q, is bisected by the line joining O l to the mid-point of P l Q l . (Refer the points P lt Q iy re- spectively, to the median as common axis of x, and to the two opposite conjugate directions as axis of y, and show that P 19 Q l are then corresponding points.) Prob. 2. Prove that the measure of a circular sector is equal to the radian measure of its angle. Prob. 3. Find the measure of an elliptic quadrant, and of the sector included by conjugate radii. ART. 4. CHARACTERISTIC RATIOS OF SECTORIAL MEASURES. Let Aftf^ = S 1 be any sector of a central conic; draw P,M^ ordinate to O l A lt i.e. parallel to the tangent at A t ; let O l M l = x lt Mf^ = jj>, , O l A l =,, and the conjugate radius O l B l = b l ; then the ratios x l /a l , yjb^ are called the charac- teristic ratios of the given sectorial measure SJK r These ratios are constant both in magnitude and sign for all sectors of the same measure and species wherever these may be situ- ated (Art. 3). Hence there exists a functional relation be- tween the sectorial measure and each of its characteristic ratios. ART. 5. RATIOS EXPRESSED AS TRIANGLE-MEASURES. The triangle of a sector and its complementary triangle are measured by the two characteristic ratios. For, let the triangle Aflfi and its complementary triangle P 1 O 1 B 1 be denoted by T 19 TV; then T\ fay i sin 67, , l ~ ^a., sn 7y \b^x^ sin K l ~ ^a l b l sin FUNCTIONAL RELATIONS FOR ELLIPSE. 11 ART. 6. FUNCTIONAL RELATIONS FOR ELLIPSE. The functional relations that exist between the sectorial measure and each of its characteristic ratios are the same for all elliptic, in- Bt eluding circular, sec- tors (Art. 4). Let/*,, P % be corresponding points on an ellipse and a circle, referred o, to the conjugate di- rections O.A^ O^BI, and right angles ; let the angle 5, K n ^B^ the latter pair being at = in radian measure; then '6 _ (6) = cos = sin a hence, in the ellipse, by Art. 3, l' = cos J-, ^nzsin L. (7) Prob. 4. Given JCi = \a\; find the measure of the elliptic sector .AiOtPi. Also find its area when a^ = 4, , = 3, GJ = 60. Prob. 5. Find the characteristic ratios of an elliptic sector whose measure is %TT. Prob. 6. Write down the relation between an elliptic sector and its triangle. (See Art. 5.) ART. 7. FUNCTIONAL RELATIONS FOR HYPERBOLA. The functional relations between a sectorial measure and its characteristic ratios in the case of the hyperbola may be written in the form x \ u -i = cosh i .1 , '-i = smh _- and these express that the ratio of the two lines on the left is a certain definite function of the ratio of the two areas on the right. These functions are called by analogy the hyperbolic 12 HYPERBOLIC FUNCTIONS. cosine and the hyperbolic sine. Thus, writing u for S t /K lt the two equations x y l . = cosh u, v = sm h u (8V a, b, serve to define the hyperbolic cosine and sine of a given secto- rial measure u ; and the hyperbolic tangent, cotangent, secant, and cosecant are then defined as follows : smh u coshu tanh u = = , coth u = -r cosh u sinh u r sech u = : -, csch u = (9) cosh 11 sinh u \ The names of these functions may be read " h-cosine," "h-sine," "h-tangent," etc., or "hyper-cosine," etc. ART. 8. RELATIONS AMONG HYPERBOLIC FUNCTIONS. Among the six functions there are five independent rela- tions, so that when the numerical value of one of the functions is given, the values of the other five can be found. Four of these relations consist of the four defining equations (9). The fifth is derived from the equation of the hyperbola V_^_ ^ b? - giving cosh* u sinh 2 u = I. (10) By a combination of some of these equations other subsidi- ary relations may be obtained; thus, dividing (10) successively by cosh 2 , sinh 2 &, and applying (9), give I tanh 2 u = sech 2 u, ) (ii) coth 2 u I = csch 2 u. 3 Equations (9), (10), (11) will readily serve to express the value of any function in terms of any other. For example, when tanh u is given, coth u = - , sech u = \ tanh 2 tanh u RELATIONS BETWEEN HYPERBOLIC FUNCTIONS. 13 I tanh u cosh u = , smh u = , V I -- tanh a # V i -- tanh a # \ i tanh 2 cscn u = - tanh u t The ambiguity in the sign of the square root may usually be removed by the following considerations : The functions cosh u, sech u are always positive, because the primary char- acteristic ratio xja l is positive, since the initial line O l A l and the abscissa O l M l are similarly directed from O lt on which- ever branch of the hyperbola P l may be situated; but the func- tions sinh u, tanh u, coth u, csch u, involve the other charac- teristic ratio y l /b l , which is positive or negative according as y l and ^ have the same or opposite signs, i.e., as the measure u is positive or negative ; hence these four functions are either all positive or all negative. Thus when any one of the func- tions sinh u, tanh u, csch &, coth u, is given in magnitude and sign, there is no ambiguity in the value of any of the six hyperbolic functions ; but when either cosh u or sech u is given, there is ambiguity as to whether the other four functions shall be all positive or all negative. The hyperbolic tangent may be expressed as the ratio of two lines. For draw the tangent line AC=t\ then y x ay tanh u = <- : - = T . - b a b x ail' / x i*~ i 1 = - b .- = - b . ( I2 )0 A M The hyperbolic tangent is the measure of the triangle OAC* For OA C at t Thus the sector AOP, and the triangles AOP, FOB, AOC, are proportional to z/, sinh #, cosh , tanh u (eqs. 5, 13) ; hence sinh u > > tanh u. (14) 14 HYPERBOLIC FUNCTIONS. Prob. 7. Express all the hyperbolic functions in terms of sinh u. Given cosh u = 2, find the values of the other functions. Prob. 8. Prove from eqs. 10, u, that coshu> sinh u, coshu>i, tanh u < i, sech u < i. Prob. 9. In the figure of Art. i, let OA = 2 , OB=i, AOB = 60, and area of sector A OP = 3; find the sectorial measure, and the two characteristic ratios, in the elliptic sector, and also in the hyper- bolic sector; and find the area of the triangle A OP. (Use tables of cos, sin, cosh, sinh.) Prob. 10. Show that coth u, sech u, csch u may each be ex- pressed as the ratio of two lines, as follows: Let the tangent at P make on the conjugate axes OA, OB, intercepts OS = m, OT = n\ let the tangent at B, to the conjugate hyperbola, meet OP in R y making BR = /; then coth u = I/ a, sech u = m/a, csch u = n/b. Prob. ii. The measure of segment AMP is sinh u cosh u u. Modify this for the ellipse. Modify also eqs. 10-14, an d probs. 8, 10. v ART. 9. VARIATIONS OF THE HYPERBOLIC FUNCTIONS. ^ Since the values of the hyperbolic functions depend only on the sectorial measure, it is convenient, in tracing their vari- ations, to consider only sectors of one half of a rectangular hyperbola, whose conjugate radii are equal, and to take the principal axis OA as the common initial line of all the sectors. The sectorial measure u assumes every value from oo, through o, to -f- oo , as the terminal point P comes in from infinity on the lower branch, and passes to infinity on the upper branch; that is, as the terminal line OP swings from the lower asymptotic posi- tion y = x, to the upper one, y = x. It is here assumed, but is proved in Art. 17, that the sector AOP becomes infinite as P passes to infinity. Since the functions cosh u, sinh u, tanh u, for any position VARIATIONS OF THE HYPERBOLIC FUNCTIONS. 15 of CP, are equal to the ratios of x, y, /, to the principal radius a, it is evident from the figure that cosh o = I, sinh = 0, tanh 0=0, (15) and that as u increases towards positive infinity, cosh u, sinh u are positive and become infinite, but tanh& approaches unity as a limit ; thus cosh oo = oo , sinh oo = oo , tanh oo = i. (16) Again, as u changes from zero towards the negative side, cosh u is positive and increases from unity to infinity, but sinh u is negative and increases numerically from zero to a negative infinite, and tanh u is also negative and increases numerically from zero to negative unity ; hence cosh ( oo ) = oo , sinh { oo ) = oo , tanh ( oo ) = i. (17) For intermediate values of u the numerical values of these functions can be found from the formulas of Arts. 16, 17, and are tabulated at the end of this chapter. A general idea of their manner of variation can be obtained from the curves in Art. 25, in which the sectorial measure ?/ is represented by the abscissa, and the values of the functions cosh u, sinh u, etc., are represented by the ordinate. The relations between the functions of // and of u are evident from the definitions, as indicated above, and in Art. 8. Thus cosh ( u) = -f- cosh u, sinh ( 11) = sinh u, \ sech (#)=-{- sech , csch ( u) csch u, > (18) tanh ( u) = tanh u, coth ( u) = coth u. } Prob. 12. Trace the changes in sech u, coth u, csch u, as u passes from --oo to + oo. Show that sinh #, cosh u are infinites of the same order when u is infinite. (It will appear in Art. 17 that sinh u, cosh u are infinites of an order infinitely higher than the order of*.) Prob. 13. Applying eq. (12) to figure, page 14, prove tanh , = tan A OP. 16 HYPERBOLIC FUNCTIONS. ART. 10. ANTI-HYPERBOLIC FUNCTIONS. x y t The equations -- = cosh u, -= = sinh u, 7 = tanh u, etc., a b b may also be expressed by the inverse notation u = cosh" 1 CC- y t u =: sinh l -7, u = tanh J -T, etc., which may be read: "u is the sectorial measure whose hyperbolic cosine is the ratio x to a" etc. ; or " u is the anti-h-cosine of x/a" etc. Since there are two values of u, with opposite signs, that correspond to a given value of cosh u, it follows that if u be determined from the equation cosh u = m, where m is a given number greater than unity, u is a two-valued function of ;//. The symbol cosh' 1 m will be used to denote the positive value of u that satisfies the equation cosh u =. m. Similarly the symbol sech" 1 ;;/ will stand for the positive value of 21 that satisfies the equation sech u m. The signs of the other functions sinh' 1 ?;/, tanh' 1 */*, coth" 1 m, csch" 1 ;, are the same as the sign of m. Hence all of the anti-hyperbolic functions of real numbers are one-valued. Prob. 14. Prove the following relations: cosh" 1 m = sinh" 1 Vm* - i, sinh~ l m = cosh" 1 riff -j- i, the upper or lower sign being used according as /;/ is positive or negative. Modify these relations for sin" 1 , cos" 1 . Prob. 15. In figure, Art. i,let OA = 2, OB = i,AO = 60; find the area of the hyperbolic sector A OP, and of the segment AMP, if the abscissa of P is 3. (Find cosh" 1 from the tables for cosh.) ART. 11. FUNCTIONS OF SUMS AND DIFFERENCES. (a) To prove the difference-formulas sinh (u v) = sinh u cosh v cosh u sinh v, ) cosh ( v) = cosh u cosh v sinh u sinh v. } Let OA be any radius of a hyperbola, and let the sectors A OP, AOQ have the measures u, v\ then u -- v is the measure of the sector QOP. Let OB, OQ f be the radii conjugate to OA, OQ\ and let the coordinates of P, Q, Q be (x l , y^), (x, y), (x', y') with reference to the axes OA, OB\ then FUNCTIONS OF SUMS AND DIFFERENCES. 17 - U smh _ ) = sinh ?^LQOP = triangle QOP ^ ^ K. AT sn a)^ sn &? ^ a l l a t = sinh u cosh z; cosh u sinh z/ ; k <2' cosh (H z>) = cosh sector QOP triangle POQ' - K. K . [Art. 5, sn sn but (20) since Q, Q' are extremities of conjugate radii ; hence cosh (u v) cosh u cosh v sinh u sinh v. In the figures u is positive and v is positive or negative. Other figures may be drawn with u negative, and the language in the text will apply to all. In the case of elliptic sectors, similar figures may be drawn, and the same language will apply, except that tlie second equation of (20) will be x r /a l = therefore sin (u v) = sin u cos v cos u sin v t cos (u v) = cos u cos v -f- sin u sin v. (b) To prove the sum-formulas sinh (u -f- v) = sinh u cosh v -f- cosh u sinh v, ) cosh (u -f- z>) cosh ?/ cosh z> -f- sinh u sinh z;. ) These equations follow from (19) by changing v into zv (21) 18 HYPERBOLIC FUNCTIONS. and then for sinh ( v), cosh ( v) y writing sinh v, cosh v (Art. 9, eqs. (18)). (c) To prove that tanh (uv\ = tgg*Ltanhp ( 22 ) i tanh tanh? Writing tanh (u v) = sin ( u v ) expanding and dividing cosh (ii _ v) numerator and denominator by cosh u cosh z>, eq. (22) is ob- tained. Prob. 1 6. Given cosh u 2, cosh v 3, find cosh (u -f- z>). Prob. 17. Prove the following identities: 1. sinh 2U 2 sinh # cosh w. 2. cosh 2u cosh 2 ?/ -|- sinh a i -}- 2 sinh 2 u = 2 cosh 3 u \. 3. i -h cosh u 2 cosh 2 #, cosh & -- i = 2 sinh 2 -JT&. sinh z; cosh u i /cosh i\* 4. tanh \u - - = - - = i -j- cosh u smh u \cosh u -f- i/ . 2 tanh & T -L- tanh 2 & 5. smh 2U = - cosh aw . i tanh u i - tanh & 6. sinh 3^ 3 sinh u -j- 4 sinh 3 #, cosh 3// = 4 cosh 3 // 3 cosh w. i -f tanh \u 7. cosh u 4- smh w i tanh \u 8. (cosh u -\- sinh ?/)(cosh z; + sinh ^) = cosh (// -f v) -\- sinh (?/ -f- T). 9. Generalize (8); and show also what it becomes when u=v= . . , 10. sinh 2 .*: cos'jy -f- cosh 2 .* sinV sinhV + sin 2 /. IT. cosh" 1 *?/ cosh~ l n = cosh" 1 ^;;/;/ y (w 2 - i)( a 12. sinh l m sinh l n = sinh *\m \ T. -\- n* ny i -f- Prob. 18. What modifications of signs are required in (21), (22), in order to pass to circular functions ? Prob. 19. Modify the identities of Prob. 17 for the same purpose. ART. 12. CONVERSION FORMULAS. To prove that cosh u^-\- cosh z/ a = 2 cosh J(^+ & 2 ) cosh \(u^ ?/ 2 ), cosh , cosh 2/ 2 = 2 sinh (#, -(- ?/ Q ) sinh %(u u\ (21} sinh ?/, -f- sinh ?/. 2 = 2 sinh (#, -(- ;/,) cosh \ r u^ u 9 \ sinh u, sinh a = 2 cosh i(?/. 4- ?/) sinh ^(u ). - \ i a v i "^ J LIMITING RATIOS. 19 From the addition formulas it follows that cosh (u -\- v) -f- cosh (u v) = 2 cosh u cosh v y cosh (u -\- v) cosh (u v) = 2 sinh u sinh v, sinh ( + ^) + sinh ( v) = 2 sinh cosh z>, sinh (w + v) -- sinh (u -- v) = 2 cosh u sinh v, and then by writing u + v = u l , u v u^, u =; J(a, + #,), z/ = (?/, a ), these equations take the form required. Prob. 20. In passing to circular functions, show that the only modification to be made in the conversion formulas is in the alge- braic sign of the right-hand member of the second formula. cosh 2 + cosh AV cosh 2U -f- cosh 4^ Prob. 21. Simplify -^-. -- ; . -- , - r r - sinh 2U -f- sinh 4z; cosh 2U cosh $v Prob. 22. Prove sinh 9 ,* sinh 2 ^ = sinh (x ~{-y) sinh (x y)* Prob. 23. Simplify cosh 2 .* cosh'jy sinh 2 A: sinh 2 ^. Prob. 24. Simplify cosh 2 jc cos 2 )' -f- sinh 2 ^ si ART. 13. LIMITING RATIOS. To find the limit, as u approaches zero, of sinh u tanh u > > u u which are then indeterminate in form. By eq. (14), sinh u > u > tanh u ; and if sinh u and tanh u be successively divided by each term of these inequalities, it follows that sinh u I < - -- < cosh u, u tanh u sech u < - < i; u but when u-^o, cosh w = I, sech u i, hence lim. si" h K __ i im . tanh a _ ^ , . u = o u u db o ^ HYPERBOLIC FUNCTIONS. ART. 14. DERIVATIVES OF HYPERBOLIC FUNCTIONS. To prove that / x d[sinh u} (*) - 1 = cosh u, du (*) <*) (</) d(cosh du du u) du / x d(coth w) (/) du d[csch ) du = sinh #, = sech s # t sech u tanh w, csch 2 w, = csch u coth u. (a) Let j = sinh u, Ay = sinh ( -f~ ^) ~~ smn u = 2 cosh (2& -f~ Au} sinh sinh -p- = cosh (u -f Au Take the limit of both sides, as Au ~ o, and put Ay dy dVsinh u) lim. - - = ~r = ~ j --- lim. then Similar to (<7). lim. cosh ( -f~ JzJ) = cosh ?/, sinh (see Art. 13) du sinh u = cosh w. du ' cosh # cosh 2 u sinh 9 cosh 8 u (25) cosh* // = sech*. DERIVATIVES OF HYPERBOLIC FUNCTIONS. 21 (d) Similar to (c). */(sech u] d I sinh u (e) . = -7- \ = n~ = seen u tanh u. du au cosh u cosh u (f) Similar to (e). It thus appears that the functions sinh u, cosh u reproduce themselves in two differentiations ; and, similarly, that the circular functions sin u, cos u produce their opposites in two differentiations. In this connection it may be noted that the frequent appearance of the hyperbolic (and circular) functions in the solution of physical problems is chiefly due to the fact that they answer the question : What function has its second derivative equal to a positive (or negative) constant multiple of the function itself? (See Probs. 28-30.) An answer such as y = cosh mx is not, however, to be understood as asserting that mx is an actual sectorial measure and y its characteristic ratio ; but only that the relation between the numbers mx and y is the same as the known relation between the measure of a hyper- bolic sector and its characteristic ratio ; and that the numerical value of y could be found from a table of hyperbolic cosines. Prob. 25. Show that for circular functions the only modifica- tions reqwired are in the algebraic signs of (b), (d). Prob. 26. Show from their derivatives which of the hyperbolic and circular functions diminish as u increases. Prob. 27. Find the derivative of tanh u independently of the derivatives of sinh , cosh u. Prob. 28. Eliminate the constants by differentiation from the equation j/ = A cosh mx -\- B sinh mx, and prove that d*y'/d& = m*y. Prob. 29. Eliminate the constants from the equation y = A cos mx -f- B sin mx, and prove that d*y/dx* = my. Prob. 30. Write down the most general solutions of the differen- tial equations 22 HYPERBOLIC FUNCTIONS. ART. 15. DERIVATIVES OF ANTI-HYPERBOLIC FUNCTIONS. (*) w w w z; = ^(sinh- 1 *) ' (26) = cosh ^/ dx V^f 2 -4- i 7 ^r) _ .1 dx </(tanh~ ] x}_ i -J dx ^(coth- 1 *) I 1 dx *)_ I dx ^/(csch- 1 * Vl - x* *}- I dx sinh" 1 ;r, x Vx* + i then x = sinh w, ^r = Vi + sinh 2 </ = iT+1? </#, ^ = ^/ Vi -j- ^ a . (3) Similar to (a). (c) Let # = tanh" 1 x, then # = tanh u, dx sech 3 u du = (i tanh 2 #)d = (i x*)du, du = dx/i x\ (d) Similar to (c). dx ~ dx\ x'~ x* I W a / x Vi x* (/) Similar to (e). Prob. 31. Prove 4- EXPANSION OF HYPERBOLIC FUNCTIONS. 23 Prob. 32. Prove ,.,_,* dx j <u-i x dx a sinh =- , a; cosh = == a Vx* -\- a* a Vx* a" . x adx ~~| . x adx ^tanh- - = -. E , ^coth" 1 -= * a a x _\x<a a x - a J*> Prob. 33. Find ^(sech" 1 x) independently of cosh" 1 x. Prob. 34. When tanh" 1 x is real, prove that coth" 1 x is imagi- nary, and conversely; except when x = i. sinh~ l x cosh" 1 x Prob. 35. Evaluate , , when ,# = 00. log x log x ART. 16. EXPANSION OF HYPERBOLIC FUNCTIONS. For this purpose take Maclaurin's Theorem, /(a) = /(o) + uf(o) + y"(o) + p. /'"(o) + . . ., z - 6- and put f(u) = sinh u, f(u] = cosh u, f"(u) = sinh #,..., then /(o) = sinh = 0, /'(o) = cosh o = I, . . .; hence sinh u = u -\-u* -\ 1 u 6 + . . . ; (27) and similarly, or by differentiation, cosh u = i-l r^'H r^ 4 + (28) 2! 4! By means of these series the numerical values of sinh u, cosh u, can be computed and tabulated for successive values of the independent variable u. They are convergent for all values of u, because the ratio of the th term to the preceding is in the first case u*/(2n i)(2n 2), and in the second case u*/(2n 2}(2n 3), both of which ratios can be made less than unity by taking n large enough, no matter what value u has. Lagrange's remainder shows equivalence of function and series^ 24 HYPERBOLIC FUNCTIONS. From these series the following can be obtained by division : tanh u = u %u s -f- T 2 T w 5 4- ^feu" 1 -\- . . . , sech u = I - J" + ^ - ^u 6 + . . . , u coth = it csch a = i (29) These four developments are seldom used, as there is no observable law in the coefficients, and as the functions tanh , sech u, coth u, csch u, can be found directly from the previously computed values of cosh u, sinh u. Prob. 36. Show that these six developments can be adapted to the circular functions by changing the alternate signs. ART. 17. EXPONENTIAL EXPRESSIONS. Adding and subtracting (27), (28) give the identities cosh u 4- sinh u = i -\- u -\- u* -I -- 7 u* -4- -u* 4- . . = e u , 2\ 3! 4! cosh u sinh u I u -\ -- -u* -- -u 3 4- -u* . . = e~ u . 2\ 3! 4! hence cosh u = \(e u -f- e~ u ), sinh u = %(e u e~ u ), 1 e * _ e -n 2 r (3) tanh u - -- , sech u = --- , etc. i P + e~ H e" + e- u } The analogous exponential expressions for sin u, cos u are cos u = \e ui -\-e~ ui \ sin u = (e ui e~ ni \ (i = V i) 22 where the symbol e ui stands for the result of substituting ui for x in the exponential development This will be more fully explained in treating of complex numbers, Arts. 28, 29. EXPANSION OF ANTI-FUNCTIONS. #5 Prob. 37. Show that the properties of the hyperbolic functions could be placed on a purely algebraic basis, by starting with equa- tions (30) as their definitions ; for example, verify the identities : sinh ( u) = sinh u, cosh ( u) = cosh u, cosh 2 u sinh 2 u i , sinh (u -f- v) = sinh u cosh v ~f- cosh u sinh v 9 </ 2 (cosh mu) </*(sinh mu) L m cosh /#, - = nr sinh w#. du du Prob. 38. Prove (cosh u -f- sinh &) M = cosh mi -f- sinh nu. Prob. 39. Assuming from Art. 14 that cosh u, sinh u satisfy the differential equation d*y/du* = y, whose general solution may be written y = Ae u -j- Be~ u , where A, B are arbitrary constants ; show how to determine A,jB'm order to derive the expressions. for cosh u y sinh u, respectively. [Use eq. (15).] Prob. 40. Show how to construct a table of exponential func- tions from a table of hyperbolic sines and cosines, and vice versa. Prob. 41. Prove u = log,, (cosh u -\- sinh u}. Prob. 42. Show that the area of any hyperbolic sector is infinite when its terminal line is one of the asymptotes. Prob. 43. From the relation 2 cosh u e u -f- e~ H prove 2 M ~ 1 (cosh #) M =cosh nu + ncosh (n2)u+^n(ni) cosh (n 4)0 + . . ., and examine the last term when n is odd or even. Find also the corresponding expression for 2 n ~ l (sinh u) n . ART. 18. EXPANSION OF ANTI-FUNCTIONS. . dfsinrr 1 x) i Since -2 -= -- '- == = (I 4- dx - ----.. 2 24 246 hence, by integration, i x* i 3 x" i 3 5 x" 1 sinh- 1 * = * - -+..., (31) 23 245 2467 the integration-constant being zero, since sinh' 1 x vanishes with x. This series is convergent, and can be used in compu- 26 HYPERBOLIC FUNCTIONS. tation, only when x < I. Another series, convergent when x > i, is obtained by writing the above derivative in the form = (^ + ,)-* = , - dx } -if"] -II + III. 135 I 1 " A. 2X* 2S 26^ "' .-. sinh- 1 * = C+log *+-' l^-ll J +13 | ' (32) , 2 2X* 2 4 4X* 2 466** where C is the integration-constant, which will be shown in Art. 19 to be equal to log e 2. A development of similar form is obtained for cosh" 1 x\ for 1 x) I/ l\ 2 -- - ~* dx x + ii + 13 J. , I 3 5 , 2^ 2 2 4 ^ 2 4 6; 6 ^ '' hence cosh- 1 x= , 2 2^ 2 2 44*' 246 6x* in which 7 is again equal to log,, 2 [Art. 19, Prob. 46]. Im order that the function cosh' 1 ^ maybe real, x must not be- less than unity ; but when x exceeds unity, this series is con- vergent, hence it is always available for computation. x Again, ..., and hence tanh' 1 ^^^-) x*-\--x*-\ x 1 + ..., (34) j / From (32), (33), (34) are derived : sech" 1 x = cosh" 1 x 2.2 2.4.4 2.4.6.6 LOGARITHMIC EXPRESSION OF ANTI-FUNCTIONS. 27 i I I II ! 3 ! !35I csch-' x = smh- 1 - = +-- - + ..., X X 2 $X 2 4 $X 246 Jx" ~ 1 x* i . 3 . x* i . 3 . 5 . x* = C \ogx-\-- - ...; (36) 2.2 2.4.4 ' 2.4.6.6 coth-' x = tanh- 1 I = I + -i- -f -L -L _L + . . .. (37) x x 3*' $x b ?S Prob. 44. Show that the series for tanh" 1 x, coth" 1 x, sech" 1 x, are always available for computation. Prob. 45. Show that one or other of the two developments of the inverse hyperbolic cosecant is available. ART. 19. LOGARITHMIC EXPRESSION OF ANTI-FUNCTIONS. Let x = cosh u, then Vx* - i = sinh u\ therefore x 4- Vx* - i == cosh u -j- sinh u = e u , and u, = cosh ~ l x, = log (x -f- l/jr 2 - i). (38) Similarly, sintrttr = log (# -(- i/^r* -f- i). (39) Also sech" 1 ^ = cosh" 1 - = log It (40) x x i i _J_ i/i _i_ ^ a cscrrttr = sinh" 1 - = log - . (41) ^M >- Again, let x = tanh u = , therefore - = ^- = <? 2 *, i ;tr e * I 4- # i 4-^r ^ ^y - I ^\f"f 1~ Q f> Kl ^* JL, I ^\ Qp / J SJ \ 3 i -- x ' I x I ^f I I and coth" 1 :*; = tanh' 1 - = J log - -. (43) ^r jir i Prob. 46. Show from (38), (39), that, when x = oo, sinh"" 1 ^: log ^ ^ log 2. cosh" ^ ^ - log jc -i log 2, and hence show that the integration-constants in (32), (33) are each <equal to log 2. 28 HYPERBOLIC FUNCTIONS. Prob. 47. Derive from (42) the series for tanh' 1 .* given in (34). Prob. 48. Prove the identities: x i oc* i log.x=2tanh~ 1 - =tanh -1 -=sinh~ J (*" ^~ 1 )= x+i x +i log sec x = 2 tanh' 1 tan 2 \x; log esc x = 2 tanh" ! tan a (i^r -f- $x)\ log tan x = tanh' 1 cos 2^" sinh" 1 cot 2x = cosh" 1 esc 2x. ART. 20. THE GUDERMANIAN FUNCTION. The correspondence of sectors of the same species was dis- cussed in Arts. 1-4. It is now convenient to treat of the correspondence that may exist between sectors of different species. Two points PV P^ , on any hyperbola and ellipse, are said to correspond with reference to two pairs of conjugates O t A l9 O^B t , and O n A^ , OJB* , respectively, when xja, = a,/x v (44). and when y lt y t have the same sign. The sectors A^O.P^ A^O^P^ are then also said to correspond. Thus corresponding sectors of central conies of different species are of the same sign and have their primary characteristic ratios reciprocal. Hence there is a fixed functional relation between their re- spective measures. The elliptic sectorial measure is called the gudermanian of the corresponding hyperbolic sectorial measure, and the latter the anti-gudermanian of the former. This relation is expressed by S,/K, = gd S,/K, or v = gd u, and u = gd" 1 ^. (45) ART. 21. CIRCULAR FUNCTIONS OF GUDERMANIAN. The six hyperbolic functions of u are expressible in terms of the six circular functions of its gudermanian ; for since ^-= cosh , = cos v, (see Arts. 6, 7) a, a, in which u, v are the measures of corresponding hyperbolic and elliptic sectors, GUDERMANIAN ANGLE. 29 1 hence cosh u = sec v, [eq. (44)] sinh u = V sec 2 ^ I = tan v, tanh u = tan ^/sec v = sin z/, coth w = esc z/, sech u = cosz/, csch & = cot v. The gudermanian is sometimes useful in computation ; for instance, if sinh u be given, v can be found from a table of natural tangents, and the other circular functions of v will give the remaining hyperbolic functions of u. Other uses of this function are given in Arts. 22-26, 32-36. Prob. 49. Prove that gd u = Bec l ~ 1 (cosh u) = tan~ 1 (sinh u) = cos'^sech u) =sin~ 1 (tanh u), Prob. 50. Prove gd" 1 ^ = cosh'^sec v) = sinh" 1 (tan v) = sech" 1 (cos v) = tahh -1 (sin v). Prob. 51. Prove gd o = o, gd oo = ^TT, gd( oo) = ^7f y Prob 52. Show that gd u and gd" 1 v are odd functions of u, v. Prob. 53. From the first identity in 4, Prob. 17, derive the rela- tion tanh \u = tan \v. Prob. 54. Prove tanh" J (tan )= gd 2u y and tan -1 (tanh x) = -J gd~ ! 2^. ART. 22. GUDERMANIAN ANGLE If a circle be used instead of the ellipse of Art. 20, the gudermanian of the hyperbolic sectorial measure will be equal to the radian measure of the angle of the corresponding circular sector (see eq. (6), and Art. 3, Prob. 2). This angle will be called the gudermanian angle ; but the gudermanian function v y as above defined, is merely a number, or ratio ; and this number is equal to the radian measure of the gudermanian angle 0, which is itself usually tabulated in degree measure ; thus 6 = iSov/7r (47) :30 HYPERBOLIC FUNCTIONS. Prob. 55. Show that the gudermanian angle of u may be construct- ed as follows: Take the principal radius OA of an equilateral hyperbola, as the initial line, and OP as the terminal line, of the sector whose measure is #; from M, the foot of the ordinate of P, draw MT tangent to the circle whose diameter is the transverse axis; then AO T is the angle required.* Prob. 56. Show that the angle 9 never exceeds 90. Prob. 57. The bisector of angle AO T M bisects the sector A OP (see Prob. 13, Art. 9, and Prob. 53, Art. 21), and the line AP. (See Prob. i, Art. 3.) Prob. 58. This bisector is parallel to TP, and the points T, P are in line with the point diametrically opposite to A. Prob. 59. The tangent at P passes through the foot of the .ordinate of T, and intersects TM on the tangent at A. Prob. 60. The angle APM is half the gudermanian angle. ART. 23. DERIVATIVES OF GUDERMANIAN AND INVERSE. Let v = gd u, u = gd" 1 v, then sec v = cosh u, sec v tan vdv = sinh u du, sec vdv = du, therefore ^(gd- 1 v) sec vdv. (48) Again, dv = cos v du = sech u du, therefore ^(gd u) = sech u du. (49) Prob. 61. Differentiate: y = sinh u gd u, y = sin v -f- gd" 1 v, y = tanh u sech u -j- gd u, y = tan v sec v -\- gd" 1 v. * This angle was called by Gudermann the longitude of u, and denoted by lu. His inverse symbol was ft; thus u = H(/). (Crelle's Journal, vol. 6, 1830.) Lambert, who introduced the angle 6, named it the transcendent angle. (Hist, de 1'acad, roy de Berlin, 1761). Hotiel (Nouvelles Annales, vol. 3, 1864) called it the hyperbolic amplitude of u, and wrote it amh u, in analogy with the amplitude of an elliptic function, as shown in Prob. 62. Cayley (Elliptic Functions, 1876) made the usage uniform by attaching to the angle the name of the mathematician who had used it extensively in tabulation and in the ,heory of elliptic functions of modulus unity SERIES FOR GUDERMANIAN AND ITS INVERSE. 31 Prob. 62. Writing the "elliptic integral of the first kind" in the form p d(f) u = Vi K* sin 8 0' K being called the modulus, and the amplitude; that is, = am u, (mod. /c), show that, in the special case when K = i, u = gd~ l 0, am u = gd u, sin am u = tanh #, cos am u = sech u, tan am u = sinh u\ and that thus the elliptic functions sin am u, etc., degenerate into the hyperbolic functions, when the modulus is unity.* ART. 24. SERIES FOR GUDERMANIAN AND ITS INVERSE. Substitute for sech u, sec v in (49), (48) their expansions, Art. 16, and integrate, then gd u = u - K + jV - jfofU 1 + . . . (50) gd-'z, = v + X + Jr*' +TO</ + . . . (51) No constants of integration appear, since gd u vanishes with u, and gd~V with ^. These series are seldom used in compu- tation, as gd u is best found and tabulated by means of tables of natural tangents and hyperbolic sines, from the equation gd u = tan -1 (sinh u), and a table of the direct function can be used to furnish the numerical values of the inverse function ; or the latter can be obtained from the equation, gd" 1 ^ = sinh~'(tan v) = cosh'^sec ^). To obtain a logarithmic expression for gd~X let gd" 1 ^ = u, v gd u, * The relation gd u = am u, (mod. i), led Hoiiel to name the function gd u t the hyperbolic amplitude of u, and to write itamh u (see note, Art. 22). In this connection Cayley expressed the functions tanh u, sech u, sinh u in the form sin gd u, cos gd u, tan gd -u, and wrote them sg u, eg u, tg u, to correspond with the abbreviations sn u, en u, dn u for sin am u, cos am u, tan am u. Thus tanh u = sg u = sn u, (mod. i); etc. It is well to note that neither the elliptic nor the hyperbol'c functions received their names on account of the relation existing between them in a special case. (See foot-note, p. 7 ) 32 therefore HYPERBOLIC FUNCTIONS. sec v = cosli u, tan v = sinh sec v -j- tan v = cosh u -\- sinh & = e", I -\- sin v I cos (^TT f^ \ & cos v sin (TT -j- &, = gd l v, = log, tan gd u u = tan Prob. 63. Evaluate u gd v v _]v=o (52) Prob. 64. Prove that gd u sin u is an infinitesimal of the fifth order, when u = o. Prob. 65. Prove the relations j7T + ^= tan' 1 **, i?r iz> ^/ ART. 25. GRAPHS OF HYPERBOLIC FUNCTIONS. Drawing two rectangular axes, and laying down a series of points whose abscissas represent, on any convenient scale, suc- cessive values of the sectorial measure, and whose ordinates represent, preferably on the same scale, the corre- sponding values of the function to be plotted, the locus traced out by this series of points will be a graphical representation of the variation of the func- tion as the sectorial meas- GRAPHS OF THE HYPERBOLIC FUNCTIONS. 33 ure varies. The equations of the curves in the ordinary carte- sian notation are : Fig. Full Lines. Dotted Lines. A y = cosh x, y = sech x ; B y = sinh x, y csch x ; C y = tanh x, y coth x ; D y = gd x. Here x is written for the sectorial measure u, and y for the numerical value of cosh u, etc. It is thus to be noted that the variables x, y are numbers, or ratios, and that the equation y cosh x merely expresses that the relation between the numbers x and y is taken to be the same as the relation be- tween a sectorial measure and its characteristic ratio. The numerical values of cosh u, sinh u, tanh u are given in the tables at the end of this chapter for values of u between o and 4. For greater values they may be computed from the devel- opments of Art. 16. The curves exhibit graphically the relations : i sech u - , csch u - , coth u = ; cosh u sinh u tanh u cosh u < I, sech u > I, tanh u > I, gd u < J?r, etc. ; sinh ( ) = sinh ?/, cosh ( u) = cosh u, tanh ( u) = tanh u, gd ( u} = gd u, etc.; cosh o = I, sinh o o, tanh = 0, csch (o/ =00 , etc.; cosh ( oo ) = oo, sinh ( oo ) = 00, tanh ( 00)= i,etc. The slope of the curve y = sinh x is given by the equation dy/dx = cosh #, showing that it is always positive, and that the curve becomes more nearly vertical as x becomes infinite. Its direction of curvature is obtained from d^y/dx* sinh x, proving that the curve is concave downward when x is nega- tive, and upward when x is positive. The point of inflexion is at the origin, and the inflexional tangent bisects the angle between the axes. 34 HYPERBOLIC FUNCTIONS. The direction of curvature of the locus y = sech x is given by d*y/dx* = sech x(2 tanh 2 x i), and thus the curve is con- cave downwards or upwards \ according as 2 tanh 2 x i is \ negative or positive. The in- *** flexions occur at the points x tanh- 1 .707, = .881, y = .707 ; and the slopes of j the inflexional tangents are -i-- that x is so small as .1. The curve y = csch x is asymptotic to both axes, but approaches the axis of x more rapidly than it approaches the axis of y, for when x = 3, y is only .1, but it is not till/ 10 The curves y = csch x, y = sinh x cross at the points x = .881, y = i. Prob. 66. Find the direction of curvature, the inflexional tan- gent, and the asymptotes of the curves y = gda% y = tanh x. Prob. 67. Show that there is no inflexion-point on the curves y = cosh x, y = coth x. Prob. 68. Show that any line y = mx -j- n meets the curve y = tanh x in either three real points or one. Hence prove that the equation tanh x = mx -\- n has either three real roots or one. From the figure give an approximate solution of the equation tanh x = x i. ELEMENTARY INTEGRALS. 35 Prob. 69. Solve the equations: cosh x x -f- 2; sinh x = |^; gd ^ = x J-TT. Prob. 70. Show which of the graphs represent even functions, and which of them represent odd ones. ART. 26. ELEMENTARY INTEGRALS. The following useful indefinite integrals follow from Arts. 14, 15. 23: Hyperbolic. Circular. 1. / sinh u du = cosh u, I sin u du = cos u, 2. I cosh u du = sinh u, i cos u du = sin u, 3. /tanh ;; dfo = log cosh ti, I tan udu =^ log cos , 4. / coth u du = log sinh u, j cot & d# = log sin u, 5. f csohudu = log tanh , / esc u du = log tan , J 2 v 2 = sinh-^csch u), = -- cosh-'(csc u), 6. / sech u du = gd u, I sec u du =. gd- 1 u, C dx x r dx x 7' J ,/-T-7 i = S1Ilh ~ - * J ~^= =* = S1 ' n ~ -> x a f dx x r -dx x J 7^ ^ = C a ' J ~^T ^ = C S " a' r *& ~~~ M* r c/' " "~ -^ c*- x T dx I . ^r / ^r i , ,x r 9. / - 9 a = -tanh- -, / - v d % | ft ft fc u, -\-x* t I " / ^ * Forms 7-12 are preferable to the respective logarithmic expressions (Art. 19), on account of the close analogy with the circular forms, and also because they involve functions that are directly tabulated. This advantage appears more clearly in 13-20. 36 HYPERBOLIC FUNCTIONS. /__^J- J /* fix T , -1 = -COth- 1 -, / zrr-COt- 1 -, ^ a^ x>a a a *s a* -\- x* a a dx i x r dx i x ~ '' a / dx i .x . =-csch~ , -*- A//I* _J_ -*- a /Z /7 dx i . x [* - dx i ;r 12. / , , . , =-csch- . / .,-- =- csc- 1 - a x < a a From these fundamental integrals the following may be derived : r dx i ax-\-b 13. / x , = 7=sinh~" -. - , a positive. ac~>b \ Vox* + 2fa + c Va Vac-P i . ax -4- b = =cosh~ _ , ^positive, ac y^ VPac l ax^-b = , - cos- , a negative. /dx - I ^^r 4- b - tanh " ac \ o ac i ax -\- b coth ~ = =f - ac ac ac Thus, /%r-^- ;=-^coth- l (^-2)l= /4 -^ "4 ar ~r3 _U coth- 1 2-coth- 1 3 /* ^ n 2 - 5 -,- ^T- =-tanh- 1 (^-2) =tanh- 1 o-tanh- 1 (.5) M ^r 4^r--j _J 2 - -5494- (By interpreting these two integrals as areas, show graph- ically that the first is positive, and the second negative.) ^ 2 /x b - tanh~ *For tanh -1 (.5) interpolate between tanh (.54) = 4930, tanh (.56) = . 50*0 (see tables, pp. 64., 65); and similarly for t.mh- 1 (.",333). ELEMENTARY INTEGRALS. 37 Xb 2 X b \ / ~L i or / COth" A / - T- ; Y b a Vab V a ~ b the real form to be taken. (Put x b = z*, and apply 9, 10.) 2 or . tan Vb a p dx 2 bx 16. I - = tanh- 1 \ / -, , J (ax)Vbx Vba \ b-a 2 / b X 2 or Vba bx 2 bx coth- 1 A / , or tan - \ / - ; ; V b a' Va b \ <*& the real form to be taken. /i I i I X 2 2 a By means of a reduction-formula this integral is easily made to depend on 8. It may also be obtained by transforming the expression into hyperbolic functions by the assumption x = a cosh u, when the integral takes the form /a* / I sinh a udu= / (cosh 2u i)du = - 2 (sinh 2u 2?/) = Jtf 2 (sinh u cosh u u), which gives 17 on replacing a cosh u by x, and # sinh u by (;r 2 # 2 )i The geometrical interpretation of the result is evident, as it expresses that the area of a rectangular-hyper- bolic segment AMP is the difference between a triangle OMP and a sector OAP. 1 8. f(a* - x*?dx = -x(c? - # f )* + -a* sin" 1 -. J x 2 v 2 a /\ I i I .2T f^ a I cf^dx ^"fjtr 2 [ # 2 ^ 5 [ ^2* sinh -1 . ' 2 2 a 20. Aec 3 <pd(j) =f(i + tan 2 0)V tan = i tan 0(i + tan 2 0) 1 -f ^ sinh" 1 (tan 0) = i sec tan + J gd' 1 0. 21. / sech 3 w^w= ^ sech u tanh ^ + i gd ?/. Prob. 71. What is the geometrical interpretation of 18, 19? Prob. 72. Show thaty (tfjc 2 -{- 2^0: + ^)^ reduces to 17. 18, 19, 38 H\PERBOLIC FUNCTIONS. respectively: when a is positive, with ac < b* ; when a is negative ; and when a is positive, with ac > ^ 2 . Prob. 73. Prove / sinh u tanh u du =. sinh # gd w, /cosh & coth u du cosh & -f- log tanh . 2 Prob. 74. Integrate Prob. 75. In the parabola y 2 = 4px, if $" be the length of arc measured from the vertex, and the angle which the tangent line makes with the vertical tangent, prove that the intrinsic equation of the curve is ds/d$> = 2p sec 3 0, s = p sec tan +/gd~ 1 0. Prob. 76. The polar equation of a parabola being r = a sec 2 -2$, referred to its focus as pole, express s in terms of #. Prob. 77. Find the intrinsic equation of the curve j>/tf = cosh x/a t and of the curve y/a = log sec x/a. Prob. 78. Investigate a formula of reduction f or / cosh" .#</.#; also integrate by parts cosh" 1 .* Jx, tanh" 1 .*: dx t (sinh" 1 x)*dx\ and show that the ordinary methods of reduction for / cos wl .*sin ff .*dk: be applied to / cosh w x sinh" x dx. can ART. 27. FUNCTIONS OF COMPLEX NUMBERS. As vector quantities are of frequent occurence in Mathe- matical Physics ; and as the numerical measure of a vector in terms of a standard vector is a complex number of the form x-\-iy, in which x, y are real, and i stands for V- - i; it becomes necessary in treating of any class of functional oper- ations to consider the meaning of these operations when per- formed on such generalized numbers.* The geometrical defini- tions of cosh?/, sinh?/, given in Art. 7, being then no longer applicable, it is necessary to assign to each of the symbols *The use of vectors in electrical theory is shown in Bedell and Crehore's Alternating Currents, Chaps. XTV-XX (first published in 1892). The advantage of introducing the complex 'measures of such vectors into the differential equa- tions is shown by Steinmetz, Proc. Elec. Congress, 1893; while the additional convenience of expressing the solution in hyperbolic functions of these complex numbers is exemplified by Kennelly, Proc. American Institute Electrical Engineers, April 1895. (See below, Art. 37.) FUNCTIONS OF COMPLEX NUMBERS. 39 cosh (x -f- iy) y sinh (x -\- iy), a suitable algebraic meaning, which should be consistent with the known algebraic values of cosh x, sinh x, and include these values as a particular case when/ = o. The meanings assigned should also, if possible, be such as to permit the addition-formulas of Art. 1 1 to be made general, with all the consequences that flow from them. Such definitions are furnished by the algebraic develop- ments in Art. 16, which are convergent for all values of u, real or complex. Thus the definitions of cosh (x -f- iy\ sinh (x -f- iy) are to be cosh (x + iy) = I + (* .+ *" + j I .52) sinh (x + iy) = (x + z + (* + * 3 + . . . ^ j From these series the numerical values of cosh (x ~\- iy), sinh (x -f- iy) could be computed to any degree of approxima- tion, when x and y are given. In general the results will come out in the complex form* cosh (x -f- iy) = a -f- ib, sinh (x -f- iy) = c -j- zV/. The other functions are defined as in Art. 7, eq. (9). Prob. 79. Prove from these definitions that, whatever u may be, cosh ( u) = cosh u, sinh () = sinh #, </ </ . . -7- cosh z^ = sinh u -r-sinh = cosh u. du du ^cosh mu = m* cosh mu. sinh mu = m* sinh mu.\ du du *It is to be borne in mind that the symbols cosh, sinh, here stand for alge- braic operators which convert one number into another; or which, in the lan- guage of vector-analysis, change one vector into another, by stretching and turning. f The generalized hyperbolic functions usually present themselves in Mathe- matical Physics as the solution of the differential equation cPty/dti* = m' 2 <p, where <p, m t u are complex numbers, the measures of vector quantities. (See Art. 37.) -40 HYPERBOLIC FUNCTIONS. ART. 28. ADDITION-THEOREMS FOR COMPLEXES. The addition-theorems for cosh (u -\- v\ etc., where u, v are complex numbers, may be derived as follows. First take u, v .as real numbers, then, by Art. II, cosh (u -\- v) = cosh u cosh v + sinh u sinh v\ hence I + ( + *)'+ =i + ' + .. . This equation is true when ?/, ^ are any real numbers. It must, then, be an algebraic identity. For, compare the terms of the rth degree in the letters 21, v on each side. Those on the left are ;(#-)- ^) r ; and those on the right, when collected, form an rth-degree function which is numerically equal to the former for more than r values of u when v is constant, and for more than r values of v when u is constant. Hence the terms of the rth degree on each side are algebraically identical func- tions of u and v.* Similarly for the terms of any other degree. Thus the equation above written is an algebraic identity, and is true for all values of ?/, v, whether real or complex. Then writing for each side its symbol, it follows that cosh (u -\- v) = cosh u cosh v -f- sinh u sinh v\ (53) and by changing v into v, cosh (// -- 7') cosh u cosh v sinh u sinh v. (54) In a similar manner is found sinh (11 v) = sinh u cosh v cosh u sinh v. (55) In particular, for a complex argument, cosh (x iy) = cosh x cosh iy sinh x sinh iy, sinh (;r iy) = sinh # cosh iy cosh ^r sinh iy. * '* If two rth-degree functions of a single variable be equal for more than r values of the variable, then they are equal for all values of the variable, and are .algebraically identical." FUNCTIONS OF PURE IMAGINARIES. 41 Prob. 79. Show, by a similar process of generalization,* that if sin Uy cos Uy exp u \ be defined by their developments in powers of Uy then, whatever u may be, sin (u -}- v) = sin u cos v + cos u sin v, cos (& -{- z>) = cos ^ cos v sin # sin z>, exp (tf + 0) = exp u exp z>. Prob. 80. Prove that the following are identities: cosh 2 u sinh 2 u = i, cosh u -f- sinh # = exp #, cosh # sinh u = exp ( u\ cosh w = ^[exp u -j- exp ( )], sinh & = |[exp u exp( u)]. ART. 29. FUNCTIONS OF PURE IMAGINARIES. In the defining identities cosh u = i -) r # 2 -] r 4 + 2! 4! 1 , i 6 sinh u = u A -u A u A- . . ., 3. ^' o * j put for u the pure imaginary iy, then cosh zj/ = i j-y -| -y* - . . . = cos j, (57) * T" I . I sinh rpr = fjr -| -(ri/) 3 -| i((y) + =* sin /' (58) and, by division, tanh iy = i tan y. (59) * This method of generalization is sometimes called the principle of the " permanence of equivalence of forms." It is not, however, strictly speaking, a "principle," but a method; for, the validity of the generalization has to be demonstrated, for any particular form, by means of the principle of the alge- braic identity of polynomials enunciated in the preceding foot-note. (See Annals of Mathematics, Vol. 6, p. 8r.) f The symbol exp u stands for "exponential function of , " which is identi- cal with e* when u is real. HYPERBOLIC FUNCTIONS. These formulas serve to interchange hyperbolic and circular functions. The hyperbolic cosine of a pure imaginary is real, and the hyperbolic sine and tangent are pure imaginaries. The following table exhibits the variation of sinh u, cosh ?/,. tanh ?/, exp &, as u takes a succession of pure imaginary values. u sinh u cosh u tanh u exp u I I \iit .ft 7* i 7(i+0 $in i oo i 1 \i-rc 7* --7 7 7(i - in o i o i \i7t -.71 -7 i -7(i + * , , ^ o oo i i \in -.71 7 i - -7(i - i) 2l7t i i * In this table .7 is written for |- 4/2, = .707 .... Prob. 81. Prove the following identities : cos y = cosh iy =.-J[exp ty + exp ( />-)], = - sinh iy = .[exp iy exp ( />')], cos y -j- / sin _y cosh /y + sinh iy = exp /v, cosjy / sin jv = cosh iy sinh iy = exp ( ty), cos ty = cosh jv, sin (y = / sinh y. Prob. 82. Equating the respective real and imaginary parts on- each side of the equation cos ny -j- / sin ny = (cos y -\- i sin y) n y express cos ny in powers of cos y, sin y ; and hence derive the cor- responding expression for cosh ny. Prob. 83. Show that, in the identities (57) and (58), y may be replaced by a general complex, and hence that sinh (x i\ ) = i sin (y ^ ix) t FUNCTIONS OF X -p- ty IN THE FORM X -}- I Y. 43 cosh{.x ty) = cos (y T ix), sin (x ty) = / sinh (y ^ ix), cos (x /y) = cosh (y =F MC). Prob. 84. From the product-series for sin.* derive that for :sinh x : X*\l X' \l X~ nx--=x\i -,1(1 -77-7111 -.-ijpl..., O BDh**(l + 3)[l + ^?Kl + ART. 30. FUNCTIONS OF x-\-iy IN THE FORM By the addition-formulas, cosh (x -\- iy) cosh x cosh iy -|- sinh x sinh zjp, sinh (x -\- iy] = sinh x cosh y/ -j- cosh x sinh zy, -but cosh iy = cos jj>, sinh iy = i sin ^, hence cosh (x 4- zV) cosh ^ cos y 4- i sinh .* sin j/, ) \ I -^ / .'I ^ I f S- \ (60) sinh (.# -f- iy) = sinh ^ cos y -\-i cosh ^ sin y. ) Thus if cosh (x -\- iy) = a -f- z#, sinh (x -\-iy) = -f- z#, then ^ := cosh ^ cos y, # = sinh ^ sin y, ) , (61) <; = sinh x cos 7, a = cosh # sin y. ) From these expressions the complex tables at the end of this chapter have been computed. Writing cosh z=Z, where z x -\- iy, Z = X-\- iY', let the complex numbers z, Z be represented on Argand diagrams, in the usual way, by the points whose coordinates are (x, y), '{X, Y) ; and let the point z move parallel to the jj/-axis, on a given line x = m, then the point Z will describe an ellipse whose equation, obtained by eliminating y between the equa- tions X = cosh m cos/, F sinh m sin y, is I - T (cosh mf (sinh m) z and which, as the parameter m varies, represents a series of confocal ellipses, the /distance) between whose foci is unity. 44: HYPERBOLIC FUNCTIONS. Similarly, if the point z move parallel to the .r-axis, on a given line y = n, the point Z will describe an hyperbola whose equa- tion, obtained by eliminating the variable x from the equations- X= cosh x cos n, Y = sinh x sin n, is ^ a F a (cos #)' (sin n)" and which, as the parameter n varies, represents a series of hyperbolas corifocal with the former series of ellipses. These two systems of curves, when accurately drawn at close intervals on the Z plane, constitute a chart of the hyper- bolic cosine ; and the numerical value of cosh (m + in) can be read off at the intersection of the ellipse whose parameter is m with the hyperbola whose parameter is n* A similar chart can be drawn for sinh (x+iy), as indicated in Prob. 85. Periodicity of Hyperbolic Functions. The functions sinh u and cosh u have the pure imaginary period 2in. For sinh (u + 2in) =sinh u cos 271 +i cosh u sin 27: = sinh M, cosh (u + 2in) =cosh u cos 2n+i sinh u sin 27r = cosh u. The functions sinh u and cosh u each change sign when the argument u is increased by the half period in. For sinh (u+in) =sinh u cos n+i cosh u sin n = sinh u, cosh (u+in) = cosh u cos n+i sinh u sin n = cosh u. The function tanh u has the period in. For, it follows from the last two identities, by dividing member by member, that tanh (u+in) =tanh u. By a similar use of the addition formulas it is shown that sinh (u + tyn) =i cosh u, cosh (u + ^in) =i sinh u. By means of these periodic, half-periodic, and quarter-periodic relations, the hyperbolic functions of x+iy are easily expressible in terms of functions of x+iy, in which y' is less than JTT. * Such a chart is given by Kennelly, Proc. A. I. E. E., April 1895, and is used by him to obtain the numerical values of cosh (x + iy). sinh (x + iy), which present themselves as the measures of certain vector quantities in the theory of alternating currents. (See Art. 37.) The chart is constructed for values of x and of y between o and 1.2; but it is available for all values of y, on account of the periodicity of the functions. FUNCTIONS OF X + iy IN THE FORM X+iY. 45 The hyperbolic functions are classed in the modern function- theory of a complex variable as functions that are singly periodic with a pure imaginary period, just as the circular functions are singly periodic with a real period, and the elliptic functions are doubly periodic with both a real and a pure imaginary period. Multiple Values of Inverse Hyperbolic Functions. It fol- lows from the periodicity of the direct functions that the inverse functions sinh" 1 m and cosh" 1 m have each an indefinite number of values arranged in a series at intervals of zin. That partic- ular value of sinh" 1 m which has the coefficient of i not greater than JTT nor less than JTT is called the principal value of sinh" 1 m\ and that particular value of cosh" 1 m which has the coefficient of i not greater than it nor less than zero is called the principal value of cosh"^. When it is necessary to distinguish between the general value and the principal value the symbol of the former will be capitalized; thus Sinh"" 1 m = sinh" 1 m + 2irit, Cosh" 1 m = cosh" 1 m + 2irn, Tanh" 1 m = tanh" 1 m+irx, in which r is any integer, positive or negative. Complex Roots of Cubic Equations. It is well known that when the roots of a cubic equation are all real they are expressible in terms of circular functions. Analogous hyperbolic expressions are easily found when two of the roots are complex. Let the cubic, with second term removed, be written Consider first the case in which b has the positive sign. Let x=r sinh u, substitute, and divide by r 3 , then 3& 2C sinh U + -? smh u = -s. V" 4 Comparison with the formula sinh 3 w + J sinh u = J sinh $u 36 3 2C sinh gives r 2 whence therefore r=2&*, sinh3w=-, u = =2& sinh sinh~ 1 46 HYPERBOLIC FUNCTIONS. in which the. sign of b* is to be taken the same a's the sign of c. Now let the principal value of sinn" 1 ^-, found from the tables, be n\ then two of the imaginary values are w2wr, hence the i e ti i n i f n 2 \ three values of x are 20* smh and 20* sinni 4 -). The 3 \3 " 3 7 last two reduce to 0*(sinh A/3 cosh --). V 3 37 Next, let the coefficient of x be negative and equal to 30. It may then be shown similarly that the substitution x=r sin leads to the three solutions n n/ n , /- n \ i c 20* sin , o*( sin v 3 cos ), where n = sin" 1 TT . 3 V 3 37 & These roots are all real when c^b*. If c>$, the substitution x = r cosh u leads to the solution x = 20* cosh ( cosh" 1 77 ) * \3 #J which gives the three roots M / 11 tl\ f 20* cosh, -0*( cosh *\/J sinh J, wherein n cosh" 1 jj, in which the sign of b* is to be taken the same as the sign of c. Prob. 85. Show that the chart of cosh (x -\- ty) can be adapted to sinh (x -j- (y), by turning through a right angle; also to sin (x ~{-ty). sinh 2X 4- /"sin 2V Prob. 86. Prove the identity tanh (x + ty) = - -. cosh 2x -j- cos 2y Prob. 87. If cosh (x + /v), = a -f- /^, be written in the " modulus and amplitude" form as r(cos -f- /'sin 0), = r exp /#, then r a = a? -j- ^ a = cosh 2 x sin a 7 = cos 2 ^ sinh 8 x, tan = b/a = tanh ar tan y. Prob. 88. Find the modulus and amplitude of sinh (x -j- ty). Prob. 80. Show that the period of exp - is ia. a Prob. 90. When m is real and > i, cos" 1 m = i cosh" 1 m t 1 m = -- /cosh" 1 m. 2 When m is real and < i, cosh" 1 m = /cos" 1 m. THE CATENARY. 47 ART. 31. THE CATENARY. A flexible inextensible string is suspended from two fixed points, and takes up a position of equilibrium under the action of gravity. It is required to find the equation of the curve in which it hangs. Let w be the weight of unit length, and s the length of arc AP measured from the lowest point A ; then ws is the weight of the portion AP. This is balanced by the terminal tensions, T acting in the tangent line at P, and H in the horizontal tangent. Resolving horizontally and vertically gives T cos = H) T sin = ws, in which is the inclination of the tangent at P\ hence ws s where c is written for H/w, the length whose weight is the constant horizontal tension ; therefore dv s ds / s dx ds -7 = -. -J = <i / * + T ~~ ~7== dx c dx \l c c t/V _|_ 8 ' ^ 5 ^rj^/J # = sinh- 1 , sinh = -=, = cosh -, c c c c ax t- c which is the required equation of the catenary, referred to an axis of x drawn at a distance c below A. The following trigonometric method illustrates the use of the gudermanian : The " intrinsic equation," s = c tan 0, gives ds = c sec 2 d<k> ; hence dx, = ds cos 0, = c sec d<p ; dy, ds sin 0, = <;sec tan 0^0 ; thus xc gd -1 0, jj/ = sec 0; whence y/c = sec = sec gd ;tr/ = cosh x/c ; and ,y/ = tan gd x/c = sinh x/c. Numerical Exercise. A chain whose length is 30 feet is suspended from two points 20 feet apart in the same hori- zontal ; find the parameter c, and the depth of the lowest point. 48 HYPERBOLIC FUNCTIONS. The equation s/c = sinh x/c gives i$/c = sinh lo/c, which, by putting lo/c = z t may be written 1.5,2 = sinh .2. By exam- ining the intersection of the graphs of/ = sinh#, y = 1.5.2, it appears that the root of this equation is z = 1.6, nearly. To find a closer approximation to the root, write the equation in the form/(V) = sinh z 1.5,2 = o, then, by the tables, /(i.6o) = 2.3756 -- 2.4000 = .0244, f(i.62) = 2.4276 2.4300 = - - .0024, /(i.64) = 2.4806 -- 2.4600 = -f- .0206; whence, by interpolation, it is found that f(i.622i) = o, and z = 1.6221, 10/2 = 6.1649. The ordinate of either of the fixed points is given by the equation y/c cosh x/c = cosh lO/c = cosh 1.6221 = 2.6306, from tables; hence y =. 16.2174, and required depth of the vertex = y c = 10.0525 feet.* Prob. 91. In the above numerical problem, find the inclination of the terminal tangent to the horizon. Prob. 92. If a perpendicular MN be drawn from the foot of the ordinate to the tangent at P, prove that MN is equal to the con- stant c, and that NP is equal to the arc AP. Hence show that the locus of N is the involute of the catenary, and has the prop- erty that the length of the tangent, from the point of contact to the axis of x, is constant. (This is the characteristic property of the tractory). Prob. 93. The tension T at any point is equal to the weight of a portion of the string whose length is equal to the ordinate y of that point. Prob. 94. An arch in the form of an inverted catenary f is 30 feet wide and 10 feet high; show that the length of the arch can be 2 3O obtained from the equations cosh z z = i, 2S = sinh z. 3 z * See a similar problem in Chap. I, Art. 7. f For the theory of this form of arch, see " Arch " in the Encyclopaedia Britannica. CATENARY OF UNIFORM STRENGTH. 49 ART. 32. CATENARY OF UNIFORM STRENGTH. If the area of the normal section at any point be made proportional to the tension at that point, there will then be a constant tension per unit of area, and the tendency to break will be the same at all points. To find the equation of the curve of equilibrium under gravity, consider the equilibrium of an element PP' whose length is ds, and whose weight is gpoads, where GO is the section at P, and p the uniform density. This weight is balanced by the difference of the vertical components of the tensions at P and P', hence d( T sin 0) = gpcods, d(T cos 0) = o ; therefore T cos H, the tension at the lowest point, and T = H sec 0. Again, if G? O be the section at the lowest point, then by hypothesis OO/GO O = T/H = sec 0, and the first equation becomes Hd(szc sin 0) gpco sec ds, or cdtan = where c stands for the constant H/gpoo^ the length of string (of section o? ) whose weight is equal to the tension at the lowest point ; hence, ds = c sec 0^0, s/c = gd -1 0, the intrinsic equation of the catenary of uniform strength. Also dx = ds cos = c dcf), dy = ds sin = c tan dcf> ; hence x c<p, y = c log sec 0, and thus the Cartesian equation is y/c = log sec x/c, in which the axis of x is the tangent at the lowest point. Prob. 95. Using the same data as in Art. 3i find the parameter c and the depth of the lowest point. (The equation x/c = gd s/c gives lo/c = gd I5/V, which, by putting i$/c = z, becomes 50 HYPERBOLIC FUNCTIONS. gd z = \z. From the graph it is seen that z is nearly 1.8. If f(z) = gd z \z y then, from the tables of the gudermanian at the end of this chapter, /(i.8o) = 1.2432 1.2000 = -f- .0432, /(i. 9 o) - 1.2739 1-2667 = + .0072, /( r '9S) 1-2881 1.3000 = .0119, whence, by interpolation, z =.1.9189 and =7.8170. Again, y/c = loge sec x/c ; but x/c = IQ/C = 1.2793; and 1.2793 radians = 73 17' 55"; hence 7 = 7.8170 X -4I9H X 2.3026 = 7.5443* the required depth.) Prob. 96. Find the inclination of the terminal tangent. Prob. 97. Show that the curve has two vertical asymptotes. Prob. 98. Prove that the law of the tension T, and of the section GO, at a distance s, measured from the lowest point along the curve, is T GO S = = cosh -; H GO O c j and show that in the above numerical example the terminal section is 3.48 times the minimum section. Prob. 99. Prove that the radius of curvature is given by p = c cosh s/c. Also that the weight of the arc s is given by W H sinh s/c t in which s is measured from the vertex. ART. 33, THE ELASTIC CATENARY. An elastic string of uniform section and density in its natu- ral state is suspended from two points. Find its equation of equilibrium. Let the element do- stretch"into ds\ then, by Hooke's law, ds da(\ -\- AT"), where A is the elastic constant of the string ; hence the weight of the stretched element ds, = gpcadv, = ,gpGods/(i + AT). Accordingly, as before, d(Tsm 0) =gpa>ds/(i + AT), and T cos = H = gpooc, hence cd(tan 0) ds/(i + // sec 0), in which jj. stands for A//, the extension at the lowest point ; THE TRACTORY. 51 therefore ds = <:(sec 2 + ^ sec3 s/c = tan + ju( sec tan + gd" 1 0), [prob. 20, p. 37 which is the intrinsic equation of the curve, and reduces to that of the common catenary when JLI o. The coordinates x, y may be expressed in terms of the single parameter by put- ting dx = ds cos = <:(sec -f- /* sec 2 0)^0, dy = ds sin = ^(sec 2 -j~ yw sec 3 0) sin </0. Whence = gd" 1 -f- /* tan 0, sec 0- These equations are more convenient than the result of eliminating 0, which is somewhat complicated. /ART. 34. THE TRACTORY.* To find the equation of the curve which possesses the property that the length of the tangent from the point of con- tact to the axis of x is con- stant. Let PT, P'T' be two con- secutive tangents such that PT=P'T' = c, and let OT = t\ draw TS perpendicular to/" 7*'; then if PP' = ds, it is evident that ST' differs from ds by an infinitesimal of a higher order. Let PT make an angle with OA, the axis of y\ then (to the first order of infinitesimals) PTdcf) = TS = TT' cos 0; that is, ctfcf) = cos 0<//, / = c gd~ J 0, x = / c sin 0, c(gd~* sin 0), y = c cos 0. This is a convenient single-parameter form, which gives all *This curve is used in Schiele's anti-friction pivot (Minchin's Statics, Vol. I, p. 242) ; and in the theory of the skew circular arch, the horizontal projection of the joints being a tractory. (See "Arch," Encyclopaedia Britannica.) The equation (p = gd t/c furnishes a convenient method of plotting the curve. 52 HYPERBOLIC FUNCTIONS. values of x, y as increases from o to \n. The value of s, ex- pressed in the same form, is found from the relation ds = ST f dt sin = c tan (pd<p, s = c log, sec 0. At the point A, o, x = o, s = o, / = o, y=c. The Cartesian equation, obtained by eliminating 0, is = gd" 1 (cos" 1 -] sin (cos~ ! -\ cosh" 1 \ / 1 ?- c \ / \ c] y \ <? If & be put for //, and be taken as independent variable, = gd u, x/c = u tanh.^, y/c sech u, s/c = log cosh ?/. Prob. 100. Given / = 2C, show that = 74 35', j = 1.3249*:, ^ := .2658*:, # = 1.0360*:. At what point is t = cl Prob. 101. Show that the evolute of the tractory is the catenary. (See Prob. 92.) Prob. 102. Find the radius of curvature of the tractory in terms of ; and derive the intrinsic equation of the involute. V ^ ART. 35. THE LOXODROME. On the surface of a sphere a curve starts from the equator in a given direction and cuts all the meridians at the same angle. To find its equation in latitude-and-longitude co- ordinates : Let the loxodrome cross two consecutive meridians AM, AN in the points/*, Q\ let PR be a parallel of lati- tude ; let OM= x, MP=.y, MN = dx, RQ dy, all in radian measure ; and let the angle MOP=RPQ = a\ then tan a = RQ/PR, but PR = MN cos MP* hence dx tan a = dy sec y, and x tan a = gd' 1 y, there being no integration-constant since y vanishes with x ; thus the re- quired equation is y = gd (x tan a). * Jones, Trigonometry (Ithaca, 1890), p. 185. COMBINED FLEXURE AND TENSION. 53 To find the length of the arc OP: Integrate the equation ds = dy esc a, whence s = y esc <x. To illustrate numerically, suppose a ship sails northeast, from a point on the equator, until her difference of longitude is 45, find her latitude and distance : Here tan a = I, andjj> = gd x = gd \n = gd (.7854) = .7152 radians: s = y\/2 = 1.0114 radii. The latitude in degrees is 40.980. If the ship set out from latitude y lt the formula must be modified as follows : Integrating the above differential equa- tion between the limits (#,, y t ) and (x^y^ gives (x, - O tan a = gd-> 2 - gd->,; hence gd~'j/ 2 = gd" 1 ^ + (x* x^) tan a, from which the final latitude can be found when the initial latitude and the differ- ence of longitude are given. The distance sailed is equal to {y* ~" Ji) csc a radii, a radius being 60 X i8o/?r nautical miles. Mercator's Chart. In this projection the meridians are parallel straight lines, and the loxodrome becomes the straight line y' = x tan a, hence the relations between the coordinates of corresponding points on the plane and sphere are x' = x, y' = gd~ l y. Thus the latitude y is magnified into gd ~ l y, which is tabulated under the name of "meridional part for latitude y " ; the values of y and of y' being given in minutes. A chart constructed accurately from the tables can be used to furnish graphical solutions of problems like the one proposed above. Prob. 103. Find the distance on a rhumb line between the points (30 N, 20 E) and (30 S, 40 E). ART. 36. COMBINED FLEXURE AND TENSION. A beam that is built-in at one end carries a load P at the other, and is also subjected to a horizontal tensile force Q ap- plied at the same point; to find the equation of the curve assumed by its neutral surface : Let x> y be any point of the 54 HYPERBOLIC FUNCTIONS. elastic curve, referred to the free end as origin, then the bend- ing moment for this point is Qy Px. Hence, with the usual notation of the theory of flexure,* which, on putting^ mx = #, a&dd*y/dx* =zd*u/dx*, becomes = tfu, dx whence u = A cosh nx -f- B sinh #, [probs. 28, 30 that is, y = mx ~\- A cosh nx + B sinh ;r. The arbitrary constants A, B are to be determined by the terminal conditions. At the free end x = o, y = o ; hence A must be zero, and y = mx -\- B sinh nx, -f- m -{- nB cosh nx ; dx but at the fixed end, x = /, and dy/dx = o, hence B = m/n cosh /, and accordingly w sinh nx y =. mx -- -: 7. n cosh nl To obtain the deflection of the loaded end, find the ordinate of the fixed end by putting x = /, giving deflection = m(l -- tanh nl\ n ' Prob. 104. Compute the deflection of a cast-iron beam, 2X2 inches section, and 6 feet span, built-in at one end and carrying a load of 100 pounds at the other end, the beam being subjected to a horizontal tension of 8000 pounds. [In this case / = 4/3, E = 15 X io 6 , Q = 8000, P = 100 ; hence n = 1/50, m = 1/80, deflection = ^(72 50 tanh 1.44) = -gV(7 2 44-69) = -34i inches.] * Merriman, Mechanics of Materials (New York, 1895), pp. 70-77, 267-269. ALTERNATING CURRENTS. 55' Prob. 105. If the load be uniformly distributed over the beam r say w per linear unit, prove that the differential equation is 73 or ~ = 2JH and that the solution isy = A cosh # + ^? sinh nx-\-mx iJ \ ^. n Show also how to determine the arbitrary constants. ART. 37. ALTERNATING CURRENTS.* In the general problem treated the cable or wire is regarded as having resistance, distributed capacity, self-induction, and leakage ; although some of these may be zero in special cases. The line will also be considered to feed into a receiver circuit of any description ; and the general solution will in- clude the particular cases in which the receiving end is either grounded or insulated. The electromotive force may, without loss of generality, be taken as a simple harmonic function of the time, because any periodic function can be expressed in a Fourier series of simple harmonics.f The E.M.F. and the current, which may differ in phase by any angle, will be supposed to have given values at the terminals of the receiver circuit ; and the problem then is to determine the E.M.F, and current that must be kept up at the generator terminals ; and also to express the values of these quantities at any inter- mediate point, distant x from the receiving end ; the four line-constants being supposed known, viz.: R = resistance, in ohms per mile, L = coefficient of self-induction, in henrys per mile, C = capacity, in farads per mile, G = coefficient of leakage, in mhos per mile.J It is shown in standard works that if any simple harmonic * See references in foot-note Art. 27. f Chapter V, Art. 8. \ Kennelly denotes these constants by r, /, c, g. Steinmetz writes s for aoLy K for ooC, Q for G, and he uses C*for current. Thomson and Tait, Natural Philosophy, Vol, I. p. 40; Rayleigh, Theory of Sound, Vol. I. p. 20; Bedell and Crehore, Alternating Currents, p. 214. 56 HYPERBOLIC FUNCTIONS. function a sin (cot -)- 6) be represented by a vector of length a and angle 6, then two simple harmonics of the same period 27T/C0, but having different values of the phase-angle 6, can be combined by adding their representative vectors. Now the E.M.F. and the current at any point of the circuit, distant x from the receiving end, are of the form e = e l sin (cot -f- 0), i = i l sin (c&t + 0'), (64) in which the maximum values e lt t lt and the phase-angles 0, 6', are all functions of x. These simple harmonics will be repre- sented by the vectors *,/0, iJO' ; whose numerical measures are the complexes *, (cos 6 -\-jsin &)*, z, (cos 6' -\- j sin 0'), which will be denoted by e, L The relations between e and z may be obtained from the ordinary equations f di de de di for, since de/dt = coe } cos (&tf -{- 0) = ooe l sin (&?/ -f- 6 -j- |-TT), then dk/dT/ will be represented by the vector coe 1 /0 -\-^n ; and di/dx by the sum of the two vectors Ge l /0, Ccoe^ /& + i 7r 5 whose numerical measures are the complexes Ge,jooCe; and similarly for de/dx in the second equation ; thus the relations between the complexes /, F are *In electrical theory the symbol j is used, instead of z, for |/ I. f Bedell and Crehore, Alternating Currents, p. 181. The sign of dx is changed, because x is measured from the receiving end. The coefficient of leakage, G, is usually taken zero, but is here retained for generality and sym- metry. ^ These relations have the advantage of not involving the time. Steinmetz derives them from first principles without using the variable t. For instance, he regards R -\-jooL as a generalized resistance-coefficient, which, when applied to i, gives an E.M.F., part of which is in phase with i, and part in quadrature with i. Kennelly calls R -\- j'ooL the conductor impedance; and G -f- jc&C the dielectric admittance; the reciprocal of which is the dielectric impedance. ALTERNATING CURRENTS. 57 Differentiating and substituting give (67) and thus e, i are similar functions of x, to be distinguished only by their terminal values. It is now convenient to define two constants ;/z, m l by the equations* m* = (R +ja>L)(G +jot>Q 9 m, = m/(G +jot>C) ; (68) and the differential equations may then be written nri > (69) the solutions of which are f e = A cosh mx -f- B sinh mx, i = A' cosh mx -\- B' sinh mx, wherein only two of the four constants are arbitrary ; for sub- stituting in either of the equations (66), and equating coeffi- cients, give (G+jct>C)A whence B' = A/m lt A' B/m,. Next let the assigned terminal values of e, z, at the receiver, be denoted by E, /; then puttings = o gives E = A, f= A', whence B = mj, B' = E/m l ; and thus the general solution is e = E cosh mx -\- mj sinh mx, '. (70) z = / cosh mx -j- m E sinh mx. * The complex constants m, m\ , are written z, y by Kennelly; and the variable length x is written Z 2 . Steinmetz writes v for m. \ See Art. 14, Probs 28-30; and Art. 27, foot-note. 58 HYPERBOLIC FUNCTIONS. If desired, these expressions could be thrown into the ordi- nary complex form X -\-jY, X' -\-jY r , by putting for the let- ters their complex values, and applying the addition-theorems- for the hyperbolic sine and cosine. The quantities X, Y, X 1 ', Y' would then be expressed as functions of x ; and the repre sentative vectors of e, i y would be e^/6, i l /&', where e* = X 3 -|- F 2 , *; = X" + Y'\ tan 6 = Y/X, For purposes of numerical computation, however, the for- mulas (70) are the most convenient, when either a chart,* or a; table,f of cosh &, sinh u, is available, for complex values of u.. Prob. io6.J Given the four line-constants: R= 2 ohms per mile, L = 20 millihenrys per mile, C= 1/2 microfarad per mile, G = o> and given GO, the angular velocity of E.M.F. to be 2000 radians, per second; then ooL = 40 ohms, conductor reactance per mile; It -{-j'ooL = 2 -f- 407 ohms, conductor impedance per mile; coC = .001 mho, dielectric susceptance per mile; G -\-j(&C = .00 1/ mho, dielectric admittance per mile; (G-\-jooC)~ l = looq/ohms, dielectric impedance per mile; m * = (R-\-jooL}(G -\-jGoC) .04 + .0027, which is the^ measure of .04005 /i77 8'; therefore m = measure of .2001 /88 34' = .0050 -f- .2000;', an ab- stract coefficient per mile, of dimensions [length]' 1 ,. m^ m/(G -\-jcoC) = 200 5; ohms. Next let the assigned terminal conditions at the receiver be:: /= o (line insulated); and E = 1000 volts, whose phase may be taken, as the standard (or zero) phase; then at any distance x, by (70),, E e = E cosh nix. i -= sinh mx, m, in which mx is an abstract complex. Suppose it is required to find the E.M.F. and current that must be kept up at a generator 100 miles away; then * Art. 30, foot-note. f See Table II. J The data for this example are taken from Kennelly's article (1. c., P- 38). ALTERNATING CURRENTS. 59 e = 1000 cosh (.5 -|- 207), i = 200(40 7)"' sinh (.5 + 2O /) ibut, by Prob. 89, cosh (.5 + 207 ) = cosh (.5 + 207 67T/) = cosh (.5 -h 1.15;) = .4600 + .475/ obtained from Table II, by interpolation between cosh (.5 + iy) and cosh (.5 + I - 2 /)l hence e = 460 + 475;' = e^cos 6 +7 sin 6), where log tan log 475 log 460 = .0139, = 45 55', and l = 460 sec 9 = 661.2 volts, the required E.M.F. Similarly sinh (.5 + 207) = sinh (.5 -j- 1.157) .2126+1.0280;', and hence + /)(- 2126 + 1.028;) = 7( T 495 + 8266;') = /;(cos 0'+7'sin 6>'), where log tan 6' = 10.7427, 0' = 79 45', *', = 1495 sec 5.25 amperes, the phase and magnitude of required current. Next let it be required to find e at x = 8; then e= 1000 cosh (.04 + i.6/) = iooo/ sinh (.04+ .037), by subtracting ^TT/, and applying page 44. Interpolation be tween sinh (o + o/) and sinh (o + .i/) gives sinh (o + .037) = ooooo + . 029957. Similarly sinh (.1 -f -037) = .10004 + -0300471 Interpolation between the last two gives sinh (.04 + .037) = .04002 + .O2999/. Hence ^=7(40.02 +29.997')= 29.99+40.027 =e l (cos $+7" sin #), -where log tan = .12530, = 126 51',*?, = 29.99 sec I2 ^ 5 1 ' = 5- 01 volts. Again, let it be required to find e at x = 16; here e = iooo cosh (.08 + 3.27) = iooo cosh (.08 + .067'), l)ut cosh (o + .067) = .9970 + 07", cosh (.1 + .067) = 1.0020 + .0067; hence cosh (.08 + .067)^:1.0010 +.00487', and e= 1001+4.87 = ^(cos #+7' sin 0), where 6 = 180 17', e l = 1001 volts. Thus at a distance of about 16 miles the E.M.F. is the same as at the receiver, but in opposite 60 HYPERBOLIC FUNCTIONS. phase. Since e is proportional to cosh (.005 -j- .2j)x, the value of x for which the phase is exactly 180 is n/.2 15.7. Similarly the phase of the E.M.F. at x 7.85 is 90. There is agreement in phase at any two points whose distance apart is 31.4 miles. In conclusion take the more general terminal conditions in which the line feeds into a receiver circuit, and suppose the current is to be kept at 50 amperes, in a phase 40 in advance of the elec- tromotive force; then / 5o(cos 40 -\-j sin 40) = 38.30 -j- 32.147, and substituting the constants in (70) gives <? = 1000 cosh (.005 -f- .2/)x -f (7821 + 62367) sinh (.005 -f .2j}x = 460+ 4757 -4748+93667= -4288+98417 = <? x (cos 0+/sin 0),. where 0= 113 33', e l = 10730 volts, the E.M.F. at sending end. This is 17 times what was required when the other end was insulated. Prob. 107. If L = o, G = o, / = o ; then m = (i -f- j)n r m, = (i +./),, where n 2 aoRC/2, n? = R/zGoC; and the solution is <?, = ~~p-E /cosh 2nx -)- cos 2tix, tan = tan nx tanh nx, V 2 i. = - E I/cosh 2nx cos 2nx> tan 6' = tan nx coth nx. 2, Prob. 108. If self-induction and capacity be zero, arid the receiving end be insulated, show that the graph of the electromotive force is a catenary if G ^ o, a line if G = o. Prob. 109. Neglecting leakage and capacity, prove that the solution of equations (66) is i = 7, e = -\- (fi ~\-JGoL)Ix. Prob. no. If x be measured from the sending end, show how equations (65), (66) are to be modified; and prove that _ i _ e = jE^ cosh mx m 1 7 sinh mx, i = 7 cosh mx ~ sinh mx t . where / refer to the sending end. ART. 38. MISCELLANEOUS APPLICATIONS. 1. The length of the arc of the logarithmic curve y a* is s = M(cos\\ ?/+logtanh %u\ in which M= i/log a, sinh u y/M. 2. The length of arc of the spiral of Archimedes r = a6 is. s = <2(sinh 2u -j- 2u), where sinh u = 6. 3. In the hyperbola x*/a* y* /& = I the radius of curva- ture is p (a* sinh 2 u -f- tf cosh 2 u)l/ab ; in which u is the measure of the sector AOP, i.e. cosh u x/a, sinh it =y/b. 4. In an oblate spheroid, the superficial area of the zone MISCELLANEOUS APPLICATIONS. 61 between the equator and a parallel plane at a distance^/ is 5 = 7fd\smh 2n -f- 2u)/2e, wherein b is the axial radius, e eccen- tricity, sinh u ey/p, and/ parameter of generating ellipse. 5. The length of the arc of the parabola y = 2px, measured from the vertex of the curve, is /==^(sinh 211 -\-2ii), in which sinh u y/p tan 0, where is the inclination of the termi- nal tangent to the initial one. 6. The centre of gravity of this arc is given by 3/^r =r/ 2 (cosh 3 u - - i), 6^ly / 2 (sinh 4u 421} ; and the surface of a paraboloid of revolution is = 27t yl. 7. The moment of inertia of the same arc about its ter. minal ordinate is / = ^\xl(x 2x] + ^p 3 N~\, where /< is the mass of unit length, and // = u ^ sinh 2u \ sinh 4&-f- T V sinh 6u. 8. The centre of gravity of the arc of a catenary measured from the lowest point is given by 4/y= 2 (sinh 2u + 2u), lx '= c*(u sinh u cosh u -f- i), in which u =x/c\ and the moment of inertia of this arc about its terminal abscissa is / = /^Xir s ' n h 3^ H~ f sm h u u cos h u )- 9. Applications to the vibrations of bars are given in Ray- leigh, Theory of Sound, Vol. I, art. 170; to the torsion of prisms in Love, Elasticity, pp. 166-74; to the flow of heat and electricity in Byerly, Fourier Series, pp. 75-81 ; to wave motion in fluids in Rayleigh, Vol. I, Appendix, p. 477, and in Bassett, Hydrodynamics, arts. 120, 384; to the theory of potential in Byerly p. 135, and in Maxwell, Electricity, arts. 172-4; to Non-Euclidian geometry and many other subjects in Gunther, Hyperbelfunktionen, Chaps. V and VI. Several numerical examples are worked out in Laisant, Essai sur les fonctions hyperboliques. .0.^ HYPERBOLIC FUNCTIONS. ART. 39. EXPLANATION OF TABLES. In Table I the numerical values of the hyperbolic functions sinh w, cosh u, tanh u are tabulated for values of u increasing from o to 4 at intervals of .02. When u exceeds 4, Table IV may be used. Table II gives hyperbolic functions of complex arguments, in which cosh (x iy) = a _ ib, sinh (x iy) = c id, .and the values of a, b, c, d are tabulated for values of x .and of y ranging separately from o to 1.5 at intervals of .1. When interpolation is necessary it may be performed in three stages. For example, to find cosh (.82 -f- 1.342) : First find cosh (.82 -f- i-30 by keeping^ at i-3 and interpolating between the entries under x = .8 and x = .9 ; next find cosh (.82 -f- i-4*') by keeping^ at 1.4 and interpolating between the entries under x = .8 and x = .9, as before; then by interpolation between cosh (.82-)- 1.33) and cosh (.82 + 1-40 find cosh( .82 + i-34*) in which x is kept at .82. The table is available for all values of y, however great, by means of the formulas on page 44: sinh (x -f- 2i7i ) = sinh^r, cosh (x-\- 2i7t) = cosh x, etc. Jt does not apply when x is greater than 1.5, but this case sel- dom occurs in practice. This table can also be used as a com- plex table of circular functions, for cos (y ix] = a q= ib, sin (y ix) = d ic ; .and, moreover, the exponential function is given by .exp ( x iy) = a c i(b d), in which the signs of c and d are to be taken the same as the sign of X, and the sign of i on the right is to be the product of the signs of x and of i on the left. Table III gives the values of v= gd u, and of the guder- jnanian angle 9 = j8o v/n^ as u changes from o to I at inter- EXPLANATION OF TABLES. 63 vals of .02, from I to 2 at intervals of .05, and from 2 to 4 at intervals of .1. In Table IV are given the values of gd u, log sinh u, log cosh u, as u increases from 4 to 6 at intervals of .1, from 6 to 7 at Intervals of .2, and from 7 to 9 at intervals of .5. In the rare cases in which more extensive tables are neces- sary, reference may be made to the tables* of Gudermann, Glaisher, and Geipel and Kilgour. In the first the Guderman- ian angle (written k) is taken as the independent variable, and increases from o to 100 grades at intervals of .01, the corre- sponding value of u (written Lk] being tabulated. In the usual case, in which the table is entered with the value of &, it gives by interpolation the value of the gudermanian angle, whose circular functions would then give the hyperbolic functions of u. When u is large, this angle is so nearly right that inter- polation is not reliable. To remedy this inconvenience Gu- dermann's second table gives directly log sinh u, log cosh u, log tanh u, to nine figures, for values of u varying by .001 from 2 to 5, and by .01 from 5 to 12. Glaisher has tabulated the values of e* and *-*, to nine sig- nificant figures, as x varies by .001 from o to .1, by .01 from o to 2, by .1 from o to 10, and by I from o to 500. From these the values of cosh x, sinh x are easily obtained. Geipel and Kilgour's handbook gives the values of cosh^, sinh x, to seven figures, as x varies by .01 from o to 4. There are also extensive tables by Forti, Gronau, Vassal, Callet, and Hoiiel ; and there are four-place tables in Byerly's Fourier Series, and in Wheeler's Trigonometry. In the following tables a dash over a final digit indicates that the number has been increased. * Gudermann in Crelle's Journal, vols. 6-9, 1831-2 (published separately under the title Theorie der hyperbolischen Functionen, Berlin, 1833). Glaisher in Cambridge Phil. Trans., vol. 13, 1881. Geipel and Kilgour's Electrical Hand- book. 64 HYPERBOLIC FUNCTIONS. TABLE I. HYPERBOLIC FUNCTIONS. . sinh u. cosh u. tanh . u. sinh u. cosh u. tanh u. .00 .0000 1.0000 .0000 1.00 1.1752 1.5431 .7616 02 0200 1.0002 0200 1.02 1.2063 1.5669 7699 04 0400 1.0008 0400 1.04 1.2379 1.5913 7779 06 0600 1.0018 0599 1.06 1.2700 1.6164 7857 08 0801 1.0032 0798 1.08 1.3025 1.6421 7932 .10 .1002- 1.0050 .0997 1.10 1.3356 1.6685 .8005 12 1203 1 0072 1194 1.12 1.3693 1.6956 8076 14 1405 1.0098 1391 1.14 1.4035 1.7233 8144 16 1607 1.0128 1586 1.16 1.4382 1.7517 8210 18 1810 1.0162 1781 1.18 1.4735 1.7808 8275 .20 .2013 1.0201 .1974 1.20 1.5095 1.8107 .8337 22 2218 1.0243 2165 1.22 1.5460 1.8412 8397 24 2423 1.0289 2355 1.24 1.5831 1.8725 8455 26 2629 1.0340 2543 1.26 1.6209 1.9045 8511 28 2837 1.0395 2729 1.28 1.6593 1.9373 8565 .30 .3045 1.0453 .2913 1.30 1.6984 1.9709 .8617 32 3255 1.0516 3095 1.32 1.7381 2.0053 8668 34 3466 1.0584 3275 1.34 1.7786 2.0404 8717 36 3678 1.0655 3452 1.36 1.8198 2.0764 8764 38 3892 1.0731 3627 1.38 1.8617 2.1132 8810 .40 .4108 1.0811 .3799 1.40 1.9043 2.1509 .8854 42 4325 1.0895 3969 1.42 1.9477 2.1894 8896 44 4543 1.0984 4136 1.44 1.9919 2.2288 8937 46 4764 1.1077 4301 1.46 2.0369 2.2691 8977 48 4986 1.1174 4462 1.48 2.0827 2.3103 9015 .50 .5211 1.1276 .4621 1.50 2.1293 2.3524 .9051 52 5438 1.1383 4777 1.52 2.1768 2.3955 9087 54 5666 1.1494 4930 1.54 2.2251 2.4395 9121 56 5897 1.1609 5080 -1.56 2.2743 2.4845 9154 58 6131 1.1730 5227 1.58 2.3245 2.5305 9186 .60 .6367 1.1855 .5370 1.60 2.3756 2.5775 .9217 62 6605 1.1984 5511 1.62 2.4276 2.6255 9246 64 6846 1.2119 5649 1.64 2.4806 2.6746 9275 66 7090 1.2258 5784 1.66 2.5346 2.7247 9302 68 7336 1.2402 5915 1.68 2.5896 2.7760 9329 .70 .7586 1.2552 .6044 1.70 2.6456 2.8283 .9354 72 7838 1.2706 6169 1.72 2.7027 2.8818 9379 74 8094 1.2865 6291 1.74 2.7609 2.9364 9402 76 8353 1.3030 6411 1.76 2.8202 2.9922 9425 78 8615 1.3199 6527 1.78 2.8806 3.0492 9447 .80 .8881 1.3374 .6640 1.80 2.9422 3.1075 .9468 82 9150 1.3555 6751 1.82 3.0049 3.1669 9488 84 9423 1.3740 6858 1,84 3.0689 3.2277 9508 86 9700 1.3932 6963 1.86 3.1340 3.2897 9527 88 9981 1.4128 7064 1.88 3.2005 3.3530 9545 .90 1.0265 1.4331 .7163 1.90 3.2682 3.4177 .9562 92 1.0554 1.4539 7259 1.92 3.3372 3.4838 9579 94 1.0847 1.4753 7352 1.94 3.4075 3.5512 9595 96 1.1144 1.4973 7443 1.96 3.4792 3.6201 9611 98 1.1446 1.5199 7531 1.98 3.5523 3.6904 9626 TABLES. 65' TABLE I. HYPERBOLIC FUNCTIONS. u. sinh u. cosh u. tanh . u. sinh u. cosh w. tanh u. 2.00 3.6269 3.7622 .9640 3.00 10.0179 10.0677 .99505 2.02 3.7028 3.8355 9654 3.02 10.2212 10.2700 99524 2.04 3.7803 3.9103 9667 3.04 10.4287 10.4765 99543 2.06 3.8593 3.9867 96bO 3.06 10.6403 10.6872 99561 2.08 3.9398 4.0647 9693 3.08 10.8562 10.9022 99578 2.10 4.0219 4.1443 .9705 3.10 11.0765 11.1215 .99594 2.12 4.1056 4.2256 9716 3.12 11.3011 11.3453 9961$ 214 4.1909 4.3085 9727 3.14 11.5303 11.5786 99626 2.16 42779 4.3932 9737 3.16 11.7641 11.8065 99640 2.18 4.3666 4.4797 9748 3.18 12.0026 12.0442 99654 2.20 4.4571 4.5679 .9757 3.20 12.2459 12.2866 .99668 2.22 4.5494 4.6580 9767 3.22 12.4941 12.5340 99681 2.24 4.6434 4.7499 9776 3.24 12.7473 32.7864 99693 2.26 4.7394 4.8437 9785 3.26 13 0056 13.0440 99705 2.28 4.8372 4.9395 - 9793 3.28 13.2691 13.3067 99717 2.30 4.9370 5.0372 .9801 3.30 13.5379 13.5748 .99728 2.32 5.0387 5.1370 9809 332 13.8121 13.8483 99738 2.34 5.1425 5.2388 9816 3.34 14.0918 14.1273 99749 2.36 5.2483 5.3427 9823 3.36 14.3772 14.4120 99758 2.38 5.3562 5.4487 9830 3.38 14.6684 14.7024 99768 2.40 5.4662 5.5569 .9837 3.40 14.9654 14.9987 .99777 2.42 5.5785 5.6674 9843 3.42 15.2684 15.3011 99786 2.44 5.6929 5.7801 9849 3.44 15.5774 15.6095 99794 2.46 58097 5.8951 9855 3.46 15.8928 15.9242 99802 2.48 5.9288 6.0125 9861 3.48 16.2144 16.2453 99810 2.50 6 0502 6.1323 .9866 3.50 16.5426 16.5728 .99817 2 52 6.1741 6.2545 9871 3.52 16.8774 16.9070 99824 2.54 6.3004 6.3793 9876 3.54 17.2190 17.2480 99831 2.56 6.4293 6.5066 9881 3.56 17.5674 17.5958 99838 2.58 6.5607 6.6364 9886 3.58 17.9228 17.9507 99844 2.60 6.6947 6.7690 .9890 3.60 18.2854 18.3128 .99850 2.62 6.8315 6.9043 9895 3.62 18.6554 18.6822 99856 2.64 6.9709 7.04-23 9899 3.64 19.0328 19.0590 99862 2.66 7.1132 7.1832 9903 3.66 19.4178 19.4435 99867 2.68 7.2583 7.3268 9906 3.68 19.8106 19.8358 99872 2.70 7.4063 7.4735 .9910 3.70 20.2113 20.2360 .99877 2.72 7.5572 7.6231 9914 3.72 20.6201 20.6443 99882 2.74 7.7112 7.7758 9917 3.74 21.0371 21.0609 99887 2.76 7.8683 7.9316 9920 3.76 21.4626 21.4859 99891 2.78 8.0285 8.0905 9923 3.78 21.8966 21.9194 99896 2.80 8.1919 8.2527 .9926 3.80 22.3394 22.3618 .99900 2.82 8.3586 8.4182 9929 3.82 22.7911 22.8131 99904 2.84 8.5287 8.5871 9932 3.84 23.2520 23.2735 99907 2.86 8.7021 8.7594 9935 3.86 23.7221 23.7432 99911 2.88 8.8791 8.9352 9937 3.88 24.2018 24.2224 99915 2.90 90596 9.1146 .9940 3.90 24.6911 24.7113 .9991 & 2.92 9.2437 9.2976 9942 3.92 25 1903 25.2101 99921 2.94 9.4315 94844 9944 3.94 25.6996 25.7190 99924 2.96 9.6231 96749 9947 3.96 26 2191 26.2382 99927 2.98 98185 9.8693 9949 3.98 26.7492 26.7679 99930 66 HYPERBOLIC FUNCTIONS. TABLE II. VALUES OF COSH (x -f- iy) AND SINK (x -j- iy). X = O X = .1 y a b c d a b c d 1.0000 0000 0000 .0000 1.0050 .00000 .10017 .0000 .1 0.9950 " 0998 1.0000 01000 09967 1003 .2 0.9801 < i 1987 0.9850 01990 09817 1997 .3 0.9553 i < 2955 0.9601 02960 09570 2970 .4 .9211 <( .3894 .9257 .03901 .09226 .3914 .5 8776 4794 8820 04802 08791 4818 .6 8253 < < 5646 8295 05656 08267 5675 .7 7648 K 6442 7687 06453 07661 0474 .8 .6967 .7174 .7002 .07186 .06979 .7200 .9 6216 7833 6247 07847 06227 7872 1.0 5403 8415 5430 08429 05412 8457 1.1 4536 it ii 8912 4559 08927 04544 8957 1.2 .3624 < .9320 .3642 .09336 .03630 9367 1.8 2675 9636 2688 09652 02680 0.9684 1.4 1700 it 9854 1708 09871 01703 0.9904 1.5 0707 < < 9975 0711 09992 00709 1.0025 \7t 0000 < 1.0000 0000 10017 00000 1.0050 1/ x = .4 x = .5 y a b c d a b c d 1.0811 .0000 .4108 .0000 1.1276 .0000 .5211 .0000 .1 1.0756 0410 4087 1079 1 . 1220 0520 5185 1126 .2 1.0595 0816 4026 2148 1.1051 1025 5107 2240 .3 1.0328 1214 3924 3195 1.0773 1540 4978 3332 .4 .9957 .1600 .3783 .4210 1.0386 .2029 .4800 .4391 .5 9487 1969 3605 5183 0.9896 2498 4573 5406 .6 8922 2319 3390 6104 0.9306 2942 4301 6367 .7 8268 2646 3142 6964 0.8624 3357 3986 7264 .8 .7532 .2947 .2862 .7755 .7856 .3738 .3631 0.8089 .9 6720 3218 2553 8468 7009 4082 3239 0.8833 1.0 5841 3456 2219 9097 6093 4385 2815 0.9489 1.1 4904 3661 1863 9635 5115 4644 2364 1.0050 1.2 .3917 .3829 .1488 1.0076 .4086 .4857 .1888 1.0510 1.3 2892 3958 1099 1.0417 3016 5021 1394 1.0865 1.4 1838 4048 0698 1.0653 1917 5135 0886 1.1163 1.5 0765 4097 0291 1.0784 0798 5198 0369 1 . 1248 \Tt 0000 4108 0000 1.0811 0000 5211 0000 1.1276 TABLES. TABLE II. VALUES OF COSH (x -f- iy) AND SINH (.* -f- iy\ X = .2. x = .3 y a b c d a b c d 1.0201 .0000 .2013 .0000 1.0453 .0000 .3045 .0000 1.0150 0201 2003 1018 1.0401 0304 3030 1044 .1 0.9997 0400 1973 2027 1.0245 0605 2985 2077 .2 0.9745 0595 1923 3014 9987 0900 2909 3089 .3 .9395 .0784 .1854 .3972 .9628 .1186 .2805 .4071 .4 8952 0965 1767 4890 9174 1460 2672 5012 .5 8419 1137 1662 5760 8627 1719 2513 5903 .6 7802 1297 1540 8571 7995 1962 2329 6734 .7 .7107 .1444 .1403 .7318 .7283 .2184 .2122 .7498 .8 6341 1577 1252 7990 6498 2385 1893 8188 .9 5511 1694 1088 8584 5648 2562 1645 8796 1,0 4627 1795 0913 9091 4742 2714 1381 9316 1.1 .3696 .1877 .0730 0.9507 .3788 .2838 .1103 0.9743 1.2 2729 1940 0539 0.9829 2796 2934 0815 1.0072 1.8 1734 1984 0342 1.0052 1777 3001 0518 1.0301 1.4 0722 2008 0142 1.0175 0739 3038 0215 1.0427 1.5 0000 2013 0000 1.0201 0000 3045 0000 1.0453 \K x = .6 x = .7 y a b c d a b c d 1.1855 .0000 .6367 .0000 1.2552 .0000 .7586 .0000 1.1795 0636 6335 1183 1.2489 0757 7548 1253 .1 1.1618 1265 6240 2355 1.2301 1542 7435 2494 .2 1 . 1325 1881 6082 3503 1.1991 2242 7247 3709 .3 1.0918 .2479 .5864 .4617 1.1561 .2954 .6987 .4888 .4 1.0403 3052 5587 5684 1.1015 3637 6657 6018 .5 0.9784 3595 5255 6694 1.0359 4253 6261 7087 .6 0.9067 4101 4869 7637 0.9600 4887 5802 8086 .7 .8259 .4567 .4436 0.8504 .8745 .5442 .5285 0.9004 .8 7369 4987 3957 0.9286 7802 5942 4715 0.9832 .9 6405 5357 3440 0.9975 6782 6383 4099 1.0562 1.0 5377 5674 2888 1.0565 5693 6760 3441 1.1186 1.1 .4296 5934 .2307 1.1049 .4548 .7070 .2749 1.1699 1.2 3171 6135 1703 1.1422 3358 7309 2029 1.2094 1.3 2015 6374 1082 1,1682 2133 7475 1289 1.2369 1.4 0839 6351 0450 1.18.25 0888 7567 0537 1.2520 1.5 0000 6367 0000 1.1855 0000 7586 0000 1.2552 \1t 68 HYPERBOLIC FUNCTIONS. TABLE II. VALUES OF COSH (x -f- iy) AND SINH(.# + iy). x = .8 x = .9 y s a b c d a b c d 1.3374 .0000 .8881 .0000 1.4331 .0000 1.0265 .0000 .1 1 . 3308 0887 8837 1335 1.4259 1025 1.0214 1431 .2 1.3108 1764 8704 2657 1.4045 2039 1.0061 2847 .3 1.2776 2625 8484 3952 1.3691 3034 0.9807 4235 A 1.2319 .3458 .8180 .5208 1.3200 .3997 .9455 .5581 .5 1.1737 4258 7794 6412 1.2577 4921 9008 6871 .6 1.1038 5015 7330 7552 1.1828 5796 8472 8092 .7 1.0229 5721 6793 8616 1.0961 6613 7851 9232 .8 .9318 .6371 .6188 0.9595 .9984 .7364 .7152 1.0280 .9 8314 6957 5521 1.0476 8908 8041 6381 1.1226 1.0 7226 7472 4798 1.1254 7743 8638 5546 1.2059 1.1 6067 7915 4028 1.1919 6500 9148 4656 1.2772 1.2 .4846 .8278 .3218 1.2465 .5193 0.9568 .3720 1.3357 1.3 3578 8557 2376 1.2887 3834 0.9891 2746 1.3809 1.4 2273 8752 1510 1.3180 2436 1.0124 1745 1.4122 1.5 0946 8859 0628 1.3341 1014 1.0239 0726 1.4295 i* 0000 .8881 0000 1.3374 0000 1.0265 0000 1.4331 X = 1.2 x = 1.3 y a b c d a b c d 1.8107 .0000 1.5095 .0000 1.9709 .0000 1.6984 .0000 .1 1.8016 1507 1.5019 1808 1.9611 1696 1.6899 1968 .2 1.7746 2999 1.4794 3598 1.9316 3374 1.6645 3916 .3 1.7298 4461 1.4420 5351 1.8829 5019 1.6225 5824 .4 1.6677 .5878 1.3903 0.7051 1.8153 .6614 1.5643 0.7675 .5 1.5890 7237 1.3247 0.8681 1.7296 8142 1.4905 0.9449 .6 1.4944 8523 1.2458 1.0224 1 6267 9590 1.4017 1.1131 .7 1.3849 9724 1.1545 1.1665 1.5074 1.0941 1.2990 1.2697 .8 1.2615 1.0828 1.0517 1.2989 1.3731 1.2183 1.1833 1.4139 .9 1.1255 1.1824 0.9383 1.4183 1 2251 1.3304 1.0557 1.5439 1.0 0.9783 1.2702 0.8156 1.5236 1.0649 1.4291 0.9176 1.6585 1.1 0.8213 1.3452 0.6847 1.6137 0.8940 1.5136 0.7704 1.7565 1.2 .6561 1.4069 .5470 1.6876 .7142 1.5830 .6154 1.8370 1.3 4844 1.4544 4038 1.7447 5272 1.6365 4543 1.8991 1.4 3078 1.4875 2566 1.7843 3350 1.6737 2887 1.9422 1.5 1281 1.5057 1068 1.8061 1394 1.6941 1201 1.9660 *r 0000 1.5095 0000 1.8107 0000 1.6984 0000 1.9709 TABLES. 69 TABLE II. VALUES OF COSH (x -\- iy) AND SINK (x -f- iy.) X = I.O X = I.I I y a b c d a b c d 1.5431 .0000 1.5354 1173 1.5123 2335 1.4742 3473 1.1752 .0000 1.1693 1541 1.1518 3066 1.1227 4560 1.6685 .0000 1.6602 1333 1.6353 2654 1.5940 3946 1.3356 .0000 1.3290 1666 1.3090 3315 1.2760 4931 .1 .2 .3 1.4213 .4576 1.3542 5634 1.2736 6636 1.1802 7571 1.0824 .6009 1.0314 7398 0.9699 8718 0.8988 9941 1.5368 .5201 1.4643 6403 1.3771 7542 1.2762 8604 1.2302 0.6498 1.1721 0.7999 1.1024 0.9421 1.0216 1,0749 .4 .5 .6 .7 1.0751 0.8430 0.9592 0.9206 0.8337 0.9889 0.6999 1.0473 .8188 1 1069 7305 1.2087 6350 1.2985 5331 1.3752 1.1625 0.9581 1.0372 1.0462 0.9015 1.1239 0.7568 1.1903 .9306 1.1969 8302 1.3070 7217 1.4040 6058 1.4870 .8 .9 1.0 i.l .5592 1.0953 4128 1.1324 2623 1.1581 1092 1.1723 .4258 1.4382 3144 1.4869 1998 1.5213 0831 1.5392 .6046 1.2449 4463 1.2870 2836 1.3162 1180 1.3323 .4840 1.5551 3573 1.6077 2270 1.6442 0945 1.6643 1.2 1.3 1.4 1.5 0000 1.1752 0000 1.5431 0000 1.3356 0000 1.6685 ** y .1 .2 .3 x = 1.4 x - 1.5. a b c d a b c d 2.1509 .0000 2.1401 1901 2.1080 3783 2.0548 5628 1.9043 .0000 1.8948 2147 1.8663 4273 1.8192 6356 2.3524 .0000 23413 2126 2.3055 4230 2.2473 6292 2.1293 .0000 2.1187 2348 2.0868 4674 2.0342 6951 1.9811 0.7416 1.8876 0.9130 1.7752 1.0753 1.6451 1.2268 1.7540 0.8376 1.6712 1.0312 1.5713 1.2145 1.4565 1.3856 2.1667 0.8292 2.0644 1.0208 1.9415 1.2023 1.7992 1.3717 1.9612 0.9161 1.8686 1.1278 1.7574 1.3283 1.6286 1.5155 .4 .5 .6 .7 1.4985 1.3661 1.3370 1.4917 1.1622 1.6024 0.9756 1.6971 1.3268 1.5430 1.1838 1.6849 1.0289 1.8099 0.8638 1.9168 1.6389 1.5275 1.4623 1.6679 1.2710 1.7917 1.0671 1.8976 1.4835 1.6875 1.3236 1.8427 1.1505 1.9795 0.9659 2.0965 .8 .9 1.0 1.1 .7794 1.7749 5754 1.8349 3656 1.8766 1522 1.8996 .6900 2.0047 5094 2.0725 3237 2.1196 1347 2.1455 .8524 1.9846 6293 2.0517 3998 2.0983 1664 2.1239 .7716 2.1925 5696 2.2667 3619 2.3182 1506 2.3465 1.2 1.3 1.4 1.5 .0000 1.9043 0000 2.1509 .0000 2.1293 .0000 2.3524 \n 70 HYPERBOLIC FUNCTIONS. TABLE III. u gdu Q u gd u B u gd u Q o o 00 .0000 0.000 .60 .5669 32.483 1.50 1.1317 64.843 .02 0200 1 . 146 .62 5837 33.444 1.55 1.1525 66.034 .04 0400 2.291 .64 6003 34.395 1.60 1.1724 67.171 .06 0600 3.486 .66 6167 35 336 1.65 1.1913 68.257 .08 0799 4.579 .68 63:29 36.265 1.70 1.2094 69.294 .10 .0998 5.720 .70 .6489 37.183 1.75 1.2267 70.284 .12 1197 6.859 .72 6648 38.091 1.80 1.2432 71.228 .14 1395 7.995 .74 6804 38.987 1.85 1.2589 72.128 .16 1593 9.128 .76 6958 39.872 1.90 1.2739 72.987 .18 1790 10.258 .78 7111 40.746 1.95 1.2881 73.805 .20 .1987 11.384 .80 .7261 41.608 2.00 1.3017 74.584 .23 2183 12.505 .82 7410 42.460 2.10 1.3271 76.037 .24 2377 13.621 .84 7557 43.299 2.20 1.3501 77.354 .26 2571 14.732 .86 7702 44.128 2.30 1.3710 78.549 .28 2764 15.837 .88 7844 44.944 2.40 1.3899 79.633 .30 .2956 16.937 .90 .7985 45.750 2.50 1.4070 80.615 .32 3147 18.030 .92 8123 46.544 2.60 1.4327 81.513 .34 3336 19.116 .94 8260 47.326 2.70 1.4366 82.310 .36 3525 *80.195 .96 8394 48.097 2.80 1.4493 83.040 .38 3712 21.267 .98 8528 48.857 2.90 1.4609 83.707 .40 .3897 22.331 1.00 .8658 49.605 300 1.4713 84.301 .42 4082 23.386 1.05 8976 51.428 3.10 1.4808 84.841 .44 4264 24.434 1.10 9581 53.178 3.20 1.4894 85.33(> .46 4446 25.473 1.15 9575 54 860 3.30 1.4971 85. 7; 5 .48 4626 26.503 1.20 9857 56.476 3.40 1.5041 86.177 .50 .4804 27.524 1.25 1.0127 58.026 3.50 1.5104 86.541 .52 4980 28.535 1.30 1.0387 59.511 3.60 1.5162 86.870 .54 5155 29.537 1.35 1.0635 60.933 3.70 1.5214 87 16s .56 5328 30.529 1.40 1.0873 62.295 3.80 1.5261 87.437 .58 5500 31.511 1.45 1.1100 63.598 3.90 1.5303 87.681 TABLE IV. u gd u log sinh u log cosh u u gd u log sinh u log cosh u 4.0 1.5342 1.4360 1.4363 5.5 1.5626 2.08758 2.08760 4.1 1.5377 1.4795 1.4797 5.6 1.5634 2.13101 2.13103 4.2 1.5408 1.5229 1.5231 5.7 1.5641 2.17444 2.17445 4.3 1 5437 1.5664 1.5665 5.8 1.5648 2.21787 2.21788 4.4 1.5462 1.6098 1.6099 5.9 1.5653 2.26130 2.26131 4.5 1.5486 1.6532 1.6533 60 1.5658 2.30473 2.30474 4.6 1.5507 1 6967 1.6968 6.2 1.6867 2.39159 8.89160 4.7 1 5526 1.7401 1.7402 6.4 1.5675 2.47845 2.47H-H) 4.8 1.5543 1.7836 1.7836 6.6 1.5681 2.56531 2.56531 4.9 1.5559 1.8270 1.8270 6.8 1.5686 2.65217 2.65217 5.0 1 5573 1.8704 1.8705 7.0 1.5690 2.73903 2.73903 5.1 1.5586 1.9139 1.9139 7.5 1.5697 2.95618 3. 956 is 5.2 1.5598 1.9573 1.9573 8.0 1.5701 3.17333 8.17333 5.3 1.5608 2.0007 2.0007 8.5 1.5704 3.39047 3.39047 5.4 1 5618 2.0442 2.0442 9.0 1.5705 3.60762 3.60762 GO 1.5708 GO 00 APPENDIX. HISTORICAL AND BIBLIOGRAPHICAL. is probably the earliest suggestion of the analogy between the sector of the circle and that of the hyperbola is found in Newton's Principia (Bk. 2, prop. 8 et seq.) in connection with the solution of a dynamical problem. On the analytical side, the first hint of the modi- fied sine and cosine is seen in Roger Cotes' Harmonica Mensurarum (1722), where he suggests the possibility of modifying the expression for the area of the prolate spheroid so as to give that of the oblate one, by a certain use of the operator V i. The actual inventor of the hyperbolic trigonometry was Vincenzo Riccati, S.J. (Opuscula ad res Phys. et Math, pertinens, Bononise, 1757). He adopted the notation Sh.^>, Ch.< for the hyperbolic functions, and Sc.^>, Cc.< for the cir- cular ones. He proved the addition theorem geometrically and derived a construction for the solution of a cubic equation. Soon after, Daviet de Foncenex showed how to interchange circular and hyperbolic func- tions by the use of V i, and gave the analogue of De Moivre's theorem, the work resting more on analogy, however, than on clear definition (Reflex, sur les quant, imag., Miscel. Turin Soc., Tom. i). Johann Heinrich Lambert systematized the subject, and gave the serial devel- opments and the exponential expressions. He adopted the notation sinh u, etc., and introduced the transcendent angle, now called the gudermanian, using it in computation and in the construction of tables (1. c. page 30). The important place occupied by Gudermann in the history of the subject is indicated on page 30. The analogy of the circular and hyperbolic trigonometry naturally played a considerable part in the controversy regarding the doctrine of imaginaries, which occupied so much attention in the eighteenth cen- tury, and which gave birth to the modern theory of functions of the 72 HYPERBOLIC FUNCTIONS. complex variable. In the growth of the general complex theory, the importance of the " singly periodic functions" became still clearer, and was gradually developed by such writers as Ferroni (Magnit. expon. log. et trig., Florence, 1782); Dirksen (Organon der tran. Anal., Ber- lin, 1845); Schellbach (Die einfach. period, funkt., Crelle, 1854); Ohm (Versuch eines volk. conseq. Syst. der Math., Niirnberg, 1855); Hotiel (Theor. des quant, complex, Paris, 1870). Many other writers have helped in systematizing and tabulating these functions, and in adapting them to a variety of applications. The following works may be espe- cially mentioned: Gronau (Tafeln, 1862, Theor. und Anwend., 1865); Forti (Tavoli e teoria, 1870); Laisant (Essai, 1874); Gunther (Die Lehre . . . , 1881). The last-named work contains a very full history and bibliography with numerous applications. Professor A. G. Green- hill, in various places in his writings, has shown the importance of both the direct and inverse hyperbolic functions, and has done much to pop- ularize their use (see Diff. and Int. Calc., 1891). The following articles on fundamental conceptions should be noticed: Macfarlane, On the definitions of the trigonometric functions (Papers on Space Analysis, N. Y., 1894); Haskell, On the introduction of the notion of hyperbolic functions (Bull. N. Y. M. Soc., 1895). Attention has been called in Arts. 30 and 37 to the work of Arthur E. Kennelly in applying the hyperbolic complex theory to the plane vectors which present them- selves in the theory of alternating currents; and his chart has been described on page 44 as a useful substitute for a numerical complex table (Proc. A. I. E. E., 1895). It may be worth mentioning in this connection that the present writer's complex table in Art. 39 is believed .to be the only one of its kind for any function of the general argument x+iy. EXPONENTIAL EXPRESSIONS AS DEFINITIONS. For those who wish to start with the exponential expressions as the definitions of sinh u and cosh u, as indicated on page 25, it is here pro- posed to show how these definitions can be easily brought into direct geometrical relation with the hyperbolic sector in the form #/# = cosh S/K, y/b = sinh S/K, by making use of the identity cosh 2 u sinh 2 u= i, and the differential relations d cosh w=sinh u du, d sinh u= cosh u du, which are themselves immediate consequences of those exponential definitions. Let OA, the initial radius of the hyperbolic sector, be EXPONENTIAL EXPRESSIONS AS DEFINITIONS. 73 taken as axis of x, and its conjugate radius OB as axis of y\ let OA = a, = b, angle AOB = w, and area of triangle AOB=K, then K= sin aj. Let the coordinates of a point P on the hyperbola be x .and ;y, then x 2 /a 2 y 2 /b 2 =i. Comparison of this equation with the identity cosh 2 u sinh 2 u=i permits the two assumptions x/a=coshu and ;y/6=sinh u, wherein u is a single auxiliary variable; and it now remains to give a geometrical interpretation to u, and to prove that u=S/K, wherein 6" is the area of the sector OAP. Let the coordinates of a second point Q be x+ Ax and y+dy, then the area of the triangle POQ is, by analytic geometry, %(xdy ydx)sm to. Now the sector POQ bears to the triangle POQ a ratio whose limit is unity, hence the differential of the sector 5 may be written dS=%(x dy y dx)sm a>= \ab sin at (cosh 2 u sinh 2 u)du=K du. By integration S=Ku, hence u=S/K, the sectorial measure (p. 10); this establishes the fundamental geometrical relations x/a=cosh S/K, y/b=siuh S/K. INDEX. Addition-theorems, pages 16, 40. Admittance of dielectric, 56. Algebraic identity, 41. Alternating currents, 38, 46, 55. Ambiguity of value, 13, 16, 45. Amplitude, hyperbolic, 31. of complex number, 46. Anti-gudermanian, 28, 30, 47, 51, 52. Anti-hyperbolic functions, 16, 22, 25, 29, 35. 45- Applications, 46 et seq. Arch, 48, 51. Areas, 8, 9, 14, 36, 37, 60. Argand diagram, 43, 58. Bassett's Hydrodynamics, 61. Beams, flexure of, 54. Bedell and Crehore's alternating cur- rents, 38, 56. Byerly's Fourier Series, etc., 61, 63. Callet's Tables, 63. Capacity of conductor, 55. Catenary, 47. of uniform strength, 49. Elastic, 48. Cay ley's Elliptic Functions, 30, 31. Center of gravity, 61. Characteristic ratios, 10. Chart of hyperbolic functions, 44, 58. Mercator's, 53. Circular functions, 7, n, 14, 18, 21, 24, 29, 35. 4i, 43- of complex numbers, 39, 41, 42. of gudermanian, 28. Complementary triangles, 10. Complex numbers, 38-46. Applications of, 55-60. Tables, 62, 66. Conductor resistance and impedance, 58. Construction for gudermanian, 30. of charts, 43. of graphs, 32. Convergence, 23, 25. Conversion-formulas, 18. Corresponding points on conies, 7, 28. sectors and triangles, 9, 28. Currents, alternating, 55. Curvature, 50, 52, 60. Cotes, reference to, 71. Deflection of beams, 54. Derived functions, 20, 22, 30. Difference formula, 16. Differential equation, 21, 25, 47, 49, 51, 52, 57- Dirksen's Organon, 71. Distributed load, 55. Electromotive force, 55, 58. Elimination of constants, 21. Ellipses, chart of confocal, 43. Elliptic functions, 7, 30, 31. integrals, 7, 31. sectors, 7, 31. Equations, Differential (see). Numerical, 35, 48, 50. E volute of tractory, 52. Expansion in series, 23, 25, 31. 76 INDEX. Exponential expressions, 24, 25, 72. Ferroni, reference to, 71. Flexure of beams, 53. Foncenex, reference to, 71. Ford's Tavoli e teoria, 63, 71. Fourier series, 55, 61. Function, anti-gudermanian (see). anti-hyperbolic (see). circular (see). elliptic (see). gudermanian (see). hyperbolic, denned, n. of complex numbers, 38. of pure imaginaries, 41. of sum and difference, 16. periodic, 44. Geipel and Kilgour's Electrical Hand- book, 63. Generalization, 41. Geometrical interpretation, 37. treatment of hyperbolic functions, jetseq., 16. Glaisher's exponential tables, 63. Graphs, 32. Greenhill's Calculus, 72. Elliptic Functions, 7. Gronau's Tafeln, 63, 72. Theor. und Anwend., 72. Gudermann's notation, 30. Gudermanian, angle, 29. function, 28, 31, 34,47, 53, 63, 70. Gunther's Die Lehre, etc., 61, 71. Haskell on fundamental notions, 72. HoiiePs notation, etc., 30, 31, 71. Hyperbola, 7 et seq., 30, 37, 44, 60. Hyperbolic functions, denned, n. addition-theorems for, 16. applications of, 46 et seq. derivatives of, 20. expansions of, 23. exponential expressions for, 24. graphs of, 32. integrals involving, 35. Hyperbolic functions of complex num- bers, 38 et seq. relations among, 12. relations to gudermanian, 29. relations to circular functions, 29, 42. tables of, 64 et seq. variation of, 20. Imaginary, see complex. Impedance, 34. Integrals, 35. Interchange of hyperbolic and circular functions, 42. Interpolation, 30, 48, 50, 59, 62. Intrinsic equation, 38, 47, 49, 51. Involute of catenary, 48. of tractory, 50. Jones' Trigonometry, 52. Kennelly on alternating currents, 38, 58* Kennelly's chart, 46, 58. Laisant's Essai, etc., 61, 71. Lambert's notation, 30. place in the history, 70. Leakage of conductor, 55. Limiting ratios, 19, 23, 32. Logarithmic curve, 60. expressions, 27, 32. Love's elasticity, 61. Loxodrome, 52. Macfarlane on definitions, 72. Maxwell's Electricity, 61. Measure, defined, 8. of sector, 9 et seq* Mercator's chart, 53. Modulus, 31, 46. Moment of inertia, 61. Multiple values, 13, 16, 45. Newton, reference to, 71. Numbers, complex, 38 et seq. Ohm, reference to, 71. Operators, generalized, 39, 56. Parabola, 38, 61. Periodicity, 44, 62. INDEX. 77- Permanence of equivalence, 41. Phase angle, 56, 59. Physical problems, 21, 38, 47 et seq. Potential theory, 6l. Product-series, 43. Pure imaginary, 41. Ratios, characteristic, 10. limiting, 19. Rayleigh's Theory of Sound, 61. Reactance of conductor, 58. Reduction formula, 37, 38. Relations among functions, 12, 29, 42. Resistance of conductor, 56. Rhumb line, 53. Riccati's place in the history, 71. Schellbach, reference to, 71. Sectors of conies, 9, 28. Self-induction of conductor, 55. Series, 23, 31. Spheroid, area of oblate, 58 Spiral of Archimedes, 60. Steinmetz on alternating currents, 38,. 55- Susceptance of dielectric, 58. Tables, 43, 62. Terminal conditions, 54, 58, 60. Tractory, 48, 51. Variation of hyperbolic functions, 14^ Vassal's Tables, 63. Vectors, 38, 56. Vibrations of bars, 61. Wheeler's Trigonometry, 61. 342 M27 1906 McMahon, James Hyperbolic functions 4th ed., enl. Physical -fit Applied Scl PLEASE DO NOT REMOVE CARDS OR SLIPS FROM THIS POCKET UNIVERSITY OF TORONTO LIBRARY